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Since the balanced shaft percentage of the closing of crack remains constant but has sharp jumps at crack locations 0. At this two-point, in the unbalanced shaft the crack opening and closing are determined by the unbalance force only. At former location, the shaft bends upwards, keeping the crack closed while at the latter it bends downwards, keeping the crack open.

Percentage of the closing of crack over a full shaft rotation for different force ratios at crack axial locations a 0.

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At others crack axial location, the crack statuses undergo transition when the angular position of the crack changes. When the crack is located between 0 to 0. When the crack is located in the middle region between 0. This is also clearly visualized in Fig. If the crack is located at two side regions, the cracked unbalance shaft will experience a stiffening process corresponding to the increase in the percentage of the closing of crack and a softening process If the crack is located at middle region. Effect of unbalance force on crack breathing behavior at a 0.

The reason for this is that unbalance force does not contribute to shaft bending at 0. The crack breathing is controlled solely by the rotor weight Refer to Fig. Again, simulation results are verified by the analytical calculations as shown in Fig. But, sequences of the crack statuses are different for each crack locations, the crack starts with a fully open status at 0.

As a result, if a crack is located around these two locations, the stiffness of the unbalanced shaft would be the same as the balanced shaft. The effect of angular position of unbalance force on the crack statuses at a different axial position along the shaft length can be observed in Fig. It is clearly seen that the crack status highly depends on the angular position of unbalance force.

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This can also be clearly visualized in Fig. Therefore, the unbalance shaft is more flexible. On the other hand, the results at crack locations between 0 and 0. This phenomenon is Thus, the stiffness of an unbalanced cracked shaft for a full rotation would have a little different from a balanced cracked shaft. However, as discussed previously in Fig. Effect of crack depth ratio on crack breathing behavior is shown in Figs.

It is clear that crack statuses of unbalanced shaft strongly depend on crack depth ratio. As crack depth ratio increases the percentage of closing decreases.

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At boundary location 0. These boundary breathing behaviors are not affected by the crack depth ratio. Effect of unbalance force orientation on crack breathing behavior at a 0. Effect of crack depth ratios on crack breathing behaviour at a 0. In this paper, a large number of finite element simulations have been performed to examine the effects of crack axial position, crack angular position, crack depth ratio, unbalanced force ratio and angular position of unbalanced force on the crack breathing.

Cracked Rotors

It is found that the crack breathing strongly depends on the different combinations of these factors. In particular, the dependence of crack breathing on the crack location along shaft length is discussed in detail. Four specific crack locations along the shaft length have been identified, where the crack shows unique breathing behaviors. Further, the stiffness of the cracked shaft can change significantly from one crack location to the others. The results obtained in this work have a significant impact on the numerical calculation and prediction of dynamic response of cracked rotors as well as the development of online crack detection techniques based on vibration signals.

In a recently published paper [24], the authors demonstrated that in a Jeffcott rotor under some unbalance force orientations and magnitudes, the typical peak at one-third of resonance frequency due to a crack could disappear as the crack became fully closed during shaft rotation.

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  5. To predict the response of a rotor with a crack at this location, the result would be inaccurate if a breathing crack model is used. The existence of these special locations is not limited to the rotor used in this study. This kind of special locations should also exist in other common rotors. With the simplified crack breathing model of weight dominance balance shaft , crack breathing is independent of crack location.

    This study shows that more accurate prediction for the dynamic response and damage severity detection of cracked rotors should consider the effect of crack location on the crack breathing. Acknowledgements Mobarak is financially supported by a Ph. Crack breathing behavior of unbalanced rotor system: A Quasi-static numerical analysis Mobarak Hossain 1 , Helen Wu 2. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Stiffness of the intact shaft and single cracked shaft has been determined on experimental set up as shown in Fig. A known weight is applied on specimen.

    With the help of dial gauge indicator, deflection of shaft due to known load is measured. Shaft has been divided into 12 sections on the circumference having 30o angular interval simultaneously. The arrangement of shaft at first position is shown in Fig. The arrangement of second transverse crack is shown in Fig. In this case, depth varies in range 0 to Further, the following crack depths were considered for analysis: 4.

    This crack is generated at the distance of mm from the middle crack.

    Cracked Rotors

    Stiffness of the double cracked shaft has been evaluated through experiments and used in analytical formulation of amplitude equation. The shafts are rotated at various speeds in range rpm, which is used for dynamic investigation. It has been found from literature that probes are placed near the bearing or over the bearing for analysing shaft behaviour as per the Tong Zhou et al.

    The accelerometer is attached on the top of bearing for capturing a signal and transmitted to display monitor. Furthermore, impact hammer test has been conducted to find out static natural frequencies. This test performed through equipment OR36 maximum range 20 KHz integrate with compact real time multi-analyzer. A single accelerometer is used which mounted at mid of the shaft.

    Experiment has been conducted on the un-cracked, single cracked and multi cracked shafts. Each specimen shaft is hit by the hammer with ICP coupling at the centre of the shaft to create a force and vibration which is captured by OROS hammer and accelerometer respectively. Cross sectional view of rotating angle between crack and loading direction. Results and Discussion. Experimental results are presented in Figs Figure 6 a shows the variation in stiffness with the single crack depth variation. It shows that the stiffness variation increases with increase in crack depth. Figure 6 b shows the variation in stiffness due to second crack.

    Different positions of rotating cracked shaft are shown in Fig. The variation of stiffness of single cracked rotor and multi cracked two crack shafts at different angles are shown in Fig.

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    It is evident from the plots that the stiffness of the shaft decreases with increase in crack depth. In multi crack analysis, the change in stiffness is marginal as compared to single crack propagation. Therefore, stress concentration increases rapidly at one point in case of single crack but the effect of stress concentration at the same point is marginal due to creation of other crack.

    The peaks indicating the amplitude of acceleration are shown in Fig. These are used to find natural frequencies for cracked and un-cracked shaft. In multi crack shaft, slope of the stiffness curve decreases due to second crack as compare to single crack shaft. Thus, increase in the stress concentration is marginal near the first middle crack as compare to the middle crack depth. The natural frequencies at various crack depth have also been obtained from Fig.