Crack breathing mechanism in a cracked shaft subject to nontrivial mass unbalance

Abstract

Rotating machinery is widely used in many industrial fields and is often damaged owing to the breathing of the fatigue crack. The fatigue crack opens and closes once per revolution during shaft rotation. The breathing of the fatigue crack reduces the stiffness of the shaft and hence alters its dynamic response. It changes the vibration characteristics of the shaft. Fatigue cracks are a common occurrence in large rotor systems and can cause catastrophic failure. Detecting faults in rotating machinery before failure is the best way to avoid damage. However, a generalised method of positively identifying a fatigue crack as the cause of anomalous vibrations is not yet available. Vibration diagnostics deliver insights into the mechanical 'health' of rotating machinery in real-time when the machine is running. However, studying the vibrations of naturally occurring fatigue cracks is difficult because shafts will often either fail before, or be taken out of service once, the crack is identified. Artificially introduced cracks do not exhibit behaviour identical to that of natural ones owing to the difficulty in cutting into a shaft and leaving a slot with close to zero radius at the crack tip. Therefore, considerable efforts have been devoted to numerically modelling cracked rotors and simulating their operating conditions so that the vibrations can be studied. Numerical modelling techniques are many and varied. In the present thesis, the literature on cracked rotor dynamics is reviewed. Of the crack modelling techniques reviewed, the second area moment method is identified as having potential for improvement. The second area moment method accounts for reduction in bending stiffness of a cracked rotor. Breathing of the fatigue crack is directly related to the second area moment at the crack location. It leads to changes in one of the shaft mechanical properties, stiffness. In a shaft with a crack, the shaft stiffness will change periodically at different rotational angles. Modelling the breathing of the fatigue crack is the key step to analyse the vibration response of a cracked shaft. This breathing phenomenon must be modelled accurately to detect the crack in a rotor. However, it is not yet fully understood how partial crack closure interacts with changes in shaft stiffness, and further, with key variables of the crack detection problem. Unfortunately, almost all existing models are not applicable near the shaft critical speed, because equations of motion developed under the assumption of rotor weight dominance are no longer suitable for analysis near the critical speed. Moreover, localised reduction in stiffness is directly related to crack depth, whereas global reduction in stiffness is directly related to the crack depth and crack location along the shaft. However, researchers opt to either ignore crack location or mitigate its effects. From the literature review, it is evident that accurate modelling, which considers the influence of the crack location and the effect of the unbalance force on the crack breathing behaviour of the fatigue crack to calculate the second area moment of inertia of a cracked shaft to form the stiffness matrix, is still absent. The first topic in this research work is developing a new unbalance model""effectual bending angle""to evaluate the crack breathing response and calculate the second area moment of inertia at any crack location along the shaft length. It is developed considering the effects of unbalance force, rotor weight, rotor physical and dimensional properties and a more realistic fixed-end boundary condition. It governs the opening and closing of a shaft crack that describes the proximity of the shaft bending direction (or shaft deformation direction) relative to the crack direction. The crack breathing behaviours have been studied for every possible crack location and shaft rotation angle. The presented model identifies unique crack breathing behaviours under the influence of unbalance force and rotor physical and dimensional properties, showing the strong dependence of the breathing mechanism on the crack location. Further, the newly developed model is used to obtain the second area moment of inertia of crack cross-section closed area at any crack location along the shaft length under the unbalance force effect about the centroid. The newly developed unbalance model results are validated through 3D FEM results. This thesis finds that this analytical unbalance model captures the main features of crack breathing and is in good agreement with the 3D FEM. However, the approach adopted in this study of using the existing balance model to identify the crack breathing behaviour and the second area moment of inertia needs to be improved. In this research work, a new method is developed to determine crack breathing, which is an improvement in terms of accuracy on adopted methods. The improvement is owing to the removal of two simplifying assumptions used by previous authors, namely, that the cracked shafts will only experience symmetrical bending and the neutral axis would lie perpendicular to the bending direction, that is, always be horizontal. Both assumptions are shown to be invalid on comparison with results from a three-dimensional finite element model. The newly developed method is then used to evaluate nonlinear crack breathing behaviour under different weight""unbalance force ratios at different crack locations by examining the percentage of opening of a crack. The breathing response predicted by the developed method is validated using the three-dimensional finite element model. The results of the algorithm show a significant improvement in accuracy when compared with data from the three-dimensional finite element model of cracked rotors. The mathematical modelling of calculating the cross-section properties, namely, the second area moment and centroid location, is also improved in this research work by considering neutral axis inclination, removing the assumption of collinearity between the bending moment and neutral axis at the crack location. The newly developed equations are used to evaluate the second area moment of inertia as a function of the crack locations and shaft's angle of rotation about centroid axes. It is found to be highly dependent on crack location, similar to crack breathing behaviours. The work presented in this thesis demonstrates that a common assumption in the literature""that the effects of axial position of a crack can be neglected""is incorrect. The second topic of this research work is analysis of the crack breathing behaviour of an unbalance shaft with a more realistic transverse slant crack and elliptical crack at different crack locations along the shaft length. A three-dimensional finite element model consisting of a two-disk rotor with a crack is simulated with unbalance mass. The finite element model is simulated using Abaqus/standard. It is simulated considering the effects of unbalance force, rotor weight, rotor physical and dimensional properties and a more realistic fixed-end boundary condition. Crack breathing behaviours are visualised by the variation of the crack closed area and represented quantitatively by the percentage of the closing of the crack. Crack breathing behaviour is found to strongly depend on its axial position, angular position and depth ratios as well as unbalance force ratios and angular position of unbalance force. Compared with the balance shaft crack breathing behaviour, two different crack breathing regions along the shaft length are identified, where shaft stiffness is larger or smaller, depending on the unbalance force orientation, magnitude and crack location. However, four specific crack locations along the shaft length are identified where the crack remains fully closed or open or the same as in balance shaft crack breathing during shaft rotation under different loading conditions. The presented research results suggest that a more accurate prediction of the dynamic response of cracked rotors can be expected on considering the effects of unbalance force and individual rotor physical properties on crack breathing. The presented method and results of this research can be used to obtain the stiffness matrix of a cracked shaft element and then to study the vibration response of a cracked rotor where the rotor-weight-dominant assumption on crack breathing no longer holds.

Date of Award	2018
Original language	English