Coupling of bending and torsional vibration of a cracked Timoshenko shaft

Abstract

A transverse surface crack is known to add to the shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented by way of a 6 × 6 matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling of motion along the directions which are indicated by the off-diagonal terms. Not all motions are coupled, however. To study the coupling of torsion and shear, a 3 × 3 flexibility matrix is used which includes the appropriate terms. Due to the shear terms of the Timoshenko beam equation of the shaft, bending vibration is finally coupled to torsional vibration. This effect is the subject of this investigation, which is of particular importance in turbomachinery operation. The equations of motion of a Timoshenko beam shaft with three degrees of freedom are derived. The free vibration of the shaft and the influence of the crack on the vibrational behaviour of the shaft is studied. The relation of the eigenvalues of the system, to the crack depth and the slenderness ratio of the shaft is derived. Moreover forced vibration analysis of the cracked shaft is performed. The significant influence of the bending vibration on the torsional vibration spectrum, and vice-versa, is demonstrated. It is believed that this effect can be very useful for rotor crack identification in service, which is of importance to turbomachinery.