Dynamic Response Analysis of Rotor-Bearing Systems With Cracked Shaft

1 Literature Review Cracks occur in mechanical components due to machining imperfections, indentations resulting from collisions with solid objects, and the presence of any nonhomogeniety in the material. Such small cracks are known to propagate due to cyclic loading in rotor-bearing systems, and may eventually lead to a catastrophic failure of the rotor. The presence of a crack is known to introduce local flexibility due to the stress concentration in the vicinity of the crack tip. This consequent change in the rotor stiffness leads to considerable changes in the dynamic response characteristics of the system. Monitoring the changes in the dynamic behavior due to presence of cracks in rotating components has recently received greater attention by both engineers and dynamic analysts, 1. In this regard, vibration monitoring can be used as a means of an early crack warning, thus safeguarding against sudden failures of such very expensive rotor systems. Petroski 2 studied the time response of a simply supported cracked beam. He evaluated the deflection of the cracked beam using Fourier series approximations, wherein a simple crack model utilizing a pair of concentrated couples at the crack location is assumed. The ineffective material adjacent to the crack is modeled by an equivalent slot which size is established experimentally. The time response of a simply supported cracked beam is presented, wherein the amplitudes of vibration of the cracked beam were found to be three times greater than that of uncracked beam. However, the crack model used in reference 2 does not account for the change in the frequency spectrum of the beam due to the presence of the crack. Gounaris and Dimarogonas 3 presented a finite element of a cracked prismatic beam. They modeled the crack using the concepts of fracture mechanics by evaluating a local flexibility matrix, and then constructing the mass and stiffness matrices of the cracked beam using a consistent finite element approach. They presented results of the forced vibration of a slender cantilever beam when excited by a cyclic force at the free tip, and concluded that vibration amplitudes are affected considerably by the crack. The amplitudes of vibration of the cracked beam with a crack depth of 0.5 yielded 50 percent greater amplitudes as compared with the amplitudes of vibration of the uncracked beam, as reported in reference 3. Collins et al. 4 presented a methodology for the detection of cracks in rotating Timoshenko shafts using axial impulses as excitations. They presented the coupled equations of motion of a rotating Timoshenko shaft due to the presence of the crack. The system of differential equations of motion with cyclic axial impulses as excitations is then solved. Their proposed methodology for the detection of cracks is based on the difference of axial frequency spectrum of cracked rotating shaft, compared to uncracked shaft. Their study, however, did not include the time response or frequency response of the shaft due to any excitation in bending directions. The frequency response analysis of a nonrotating cracked shaft for coupled longitudinal-bending vibrations 5, coupled torsional-bending vibrations 6, and with a closing crack 7 has been addressed. Other investigators 8‐11 studied the frequency response and the unbalance response of rotating cracked shafts.