# Dependence of Shaft Stiffness on the Crack Location

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3 the iterative procedure seen in [13]. For a fully open crack or fully closed crack the percentage of opening of a crack, Ʌ, is equal to 100 and 0, respectively. Ʌ % 100 (12) cos 1 12 (13) Fig. 4 Crucial angles for unbalanced shaft: (a) φ 1 crack begins to close and (b) φ 2 crack become fully closed Fig. 5 Statuses of the crack breathing in the range of effectual bending angle from 0 degree to 2π The statuses of the crack at different crack locations during shaft rotation are identified using the values of effectual bending angle relative to the regions formed by φ 1 and φ 2. Fig. 5 shows the full statuses of the crack breathing in the range of bending angle from 0 degree to 2π. The percentages of opening of cracks, Ʌ, (as described in (12)) are determined by studying the transient change in closed area of the crack. A crack is the area of the crack segment at time zero as shown in Fig. 1 and A closed (t) is the closed portion of the crack segment when φ 1 φ φ 2 or 2π-φ 2 ) φ (2π-φ 1 ) (see Fig. 6). A crack can be calculated according to (13) and A closed (t) is determined using Fig. 6 Schematic diagram of the closed portion of a breathing crack III. RESULTS AND DISCUSSIONS The breathing behavior of a crack in a shaft is related directly to the stiffness of the shaft. When a crack is fully closed the shaft will have a maximum stiffness and so it will be virtually identical to an intact shaft. Similarly, when a crack is fully opened the corresponding shaft stiffness will be at a minimum. For partially open/closed crack status the shaft stiffness is between the maximum and minimum values. In order to know the influences of crack location, the analysis has considered different crack location factors, λ, (the ratio of the crack position and shaft total length, L) with different force ratios, η (the ratio of the static force and the dynamic force). A series of analyses has been done using MATLAB. In this study the shaft rotation is anticlockwise and the initial crack direction aligns with the negative Y-axis. A crack with a depth ratio of μ=0.5 and angular position of unbalance force relative to the crack direction, β=0 is chosen to perform the analysis. Throughout the paper a focus is placed on the influences of the crack location on the status of the crack. Fig. 7 describes the variation of percentage of opening of a crack along the shaft length. It is clear that crack opening and closing strongly depend on the crack location. For a balanced shaft crack opening percentage has a jump at crack locations L and L, respectively. This is caused by the change of gravitational moment direction which leads to a change of the shaft bending direction. It is seen that crack opening percentage in the balanced shaft in each of three regions divided at L and L remain unchanged. However, the variation sequence of the status of the crack in the middle region during a shaft revolution is opposite to that 322

4 when the crack is situated at side regions, as shown in Fig. 8. When the crack is located at 0.5L (middle region), the crack follows a sequential change from fully open, partially open/ closed, fully closed, partially open/closed and then fully open again. When the crack is located at 0.125L (side region), the crack follows a sequential change beginning with a fully closed status. Further, the status of the crack in a balanced shaft during a shaft rotation is symmetrical (first half same as second half of the shaft rotation). Shaft stiffness during a full shaft rotation would be the same regardless of the crack location [14]. counterpart, which indicates that the balanced shaft is stiffer than the unbalanced shaft. And when the crack is located in two side regions, the percentage of opening is clearly lower than the balanced shaft, so unbalanced shaft is stiffer than the balanced shaft. This conclusion is drawn at a specific shaft rotational angle of 135 degree. However, this conclusion holds true during a full shaft rotation, as will be discussed later. Fig. 7 Percentage of opening of a crack as a function of crack location for different force ratios where shaft rotation angle 135 Fig. 8 Statuses of a crack in the balanced shaft during a shaft rotation at crack locations (a) λ = and (b) λ=0.5 For an unbalanced shaft as shown in Fig. 7, the variation of percentage of opening of a crack along the shaft length is remarkably different from the balanced shaft. At crack locations L or L, it is independent of the force ratios, η. At former location, the crack is fully closed just like an uncracked shaft and the crack will never open during shaft rotation (see Fig. 9). The shaft will have a maximum stiffness and becomes virtually identical to an intact shaft. At the latter location L, the crack is fully open just like a shaft with a notch and will never close during shaft rotation. The shaft will have a minimum stiffness. The variation of opening percentage along shaft length can be clearly divided into three regions at 0.3L and L. In the middle region, it is obvious that the percentage of opening of the crack for the unbalanced shaft is larger than that for the balanced Fig. 9 Statuses of a crack in the unbalanced shaft during a shaft rotation at crack locations (a) λ = and (b) λ = The other interesting pair of crack locations are at 0.3L and L, where the percentages of opening of a crack for all force ratios intersect at the same value for the balanced shaft (see Fig. 7). As a result, the cracks will breathe as they would in a balanced shaft. Therefore, unbalanced shaft stiffness would be the same as the balanced shaft. If the crack is located around these two positions then the effect of unbalance force on the crack breathing behavior can be ignored. However, the variation sequence of the crack status with shaft rotation is opposite to each other at these two crack locations, as shown in Fig. 10. Fig. 10 Statuses of a crack in the unbalanced shaft during a shaft rotation at crack locations (a) λ = 0.3 and (b) λ = Figs. 11 and 12 show the statuses of a crack with different force ratios during a full shaft rotation at crack locations 0.5L and 0.125L, respectively. As the unbalance force decreases (i. e. force ratio increases) the angular range of either closed crack status or opened crack status will progressively approach those for the balanced shaft (comparing with Fig. 8), which demonstrates that the unbalanced model will be finally in agreement with a balanced model when force ratio is large 323

5 enough. This conclusion can also be drawn from percentage of opening as seen in Fig. 7. Comparing Figs. 11 and 12, it was found that the angular range of the opened crack status for a given η during a complete shaft rotation at crack location 0.5L is much wider than that at 0.125L, which is consistent with previous conclusion that the unbalanced shaft with crack at middle region is more flexible than the balanced shaft. Fig. 11 Statuses of a crack during a shaft rotation at crack location λ = 0.5 for different force ratios (a) η = 100, (b) η = 10, (c) η = 2 and (d) η = 1 Fig. 12 Statuses of a crack during a shaft rotation at crack location λ = for different force ratios (a) η = 100, (b) η = 10, (c) η = 2 and (d) η = 1 IV. CONCLUSIONS In this study, a new analytical model is developed to study the effects of crack location on the shaft stiffness. The effectual bending angle was introduced to describe the breathing behavior of a crack (open, partial open/closed, closed and percentages of opening of a crack) at different crack location and it is further used to determine the transient change in closed area of the crack to examine the percentages of opening. It was found that the transition sequence of the crack status in a balanced shaft is symmetrical about shaft rotation and is opposite to each other when the crack is in the middle region or side regions. The angular range of each crack status and the global stiffness of the balanced shaft remain unchanged regardless of crack location. Notably different crack breathing behaviors have been identified for the unbalanced shaft. At crack location L a crack is fully closed and the crack will never open. On the other hand at crack location L the crack is fully open and the crack will never close. Shaft stiffness would be maximum at the former crack location and minimum at latter location. At either crack location 0.3L or L, an unbalanced shaft is behaves completely like a balanced shaft, the stiffness would be the same as the balanced shaft. For other crack locations unbalanced shaft stiffness would be between maximum and minimum values. Unbalanced shaft stiffness variation with crack location can be divided into three regions at crack locations 0.3L and L. When the crack is located in the middle region the unbalanced shaft is more flexible than the balanced shaft and stiffer when the crack is in the two remaining regions. As the unbalance force decreases, the breathing behavior of a crack in the unbalanced shaft will gradually approach that in the balanced shaft. The developed new model can be further used to obtain the time-varying stiffness matrix of the cracked shaft element consisting of area moments of inertia and then to investigate the vibration behavior of the cracked rotor by solving the equation of motions. ACKNOWLEDGMENT The author would like to gratefully acknowledge the financial support given by the School of Computing, Engineering and Mathematics, Western Sydney University, Penrith NSW 2747 for development of this research. REFERENCES [1] S.K. Georgantzinos and N.K. Anifantis, An insight into the breathing mechanism of a crack in a rotating shaft. Journal of Sound and Vibration, : p [2] Ming, L., et al., Multi-fault diagnosis of rotor system based on differential-based empirical mode decomposition. Journal of Vibration and Control, 2013: p [3] Kulesza, Z., Dynamic behavior of cracked rotor subjected to multisine excitation. Journal of Sound and Vibration, (5): p [4] Yan, G., et al., A novel approach to detecting breathing-fatigue cracks based on dynamic characteristics. Journal of Sound and Vibration, (2): p [5] Dimarogonas, A. and C. Papadopoulos, Vibration of cracked shafts in bending. Journal of Sound and Vibration, (4): p [6] Papadopoulos, C. and A. Dimarogonas, Coupled longitudinal and bending vibrations of a rotating shaft with an open crack. Journal of sound and vibration, (1): p

6 [7] Sekhar, A.S., Crack detection through wavelet transform for a run-up rotor. Journal of Sound and Vibration, (2): p [8] Sinou, J.-J., Effects of a crack on the stability of a non-linear rotor system. International Journal of Non-Linear Mechanics, (7): p [9] Sinou, J.-J., Detection of cracks in rotor based on the 2 and 3 superharmonic frequency components and the crack unbalance interactions. Communications in Nonlinear Science and Numerical Simulation, (9): p [10] Cheng, L., et al., The influence of crack breathing and imbalance orientation angle on the characteristics of the critical speed of a cracked rotor. Journal of Sound and Vibration, (9): p [11] Bachschmid, N., P. Pennacchi, and E. Tanzi, Cracked rotors: a survey on static and dynamic behaviour including modelling and diagnosis. 2010: Springer Science & Business Media. [12] Rubio, L., et al., Quasi-static numerical study of the breathing mechanism of an elliptical crack in an unbalanced rotating shaft. Latin American Journal of Solids and Structures, : p [13] Al-Shudeifat, M.A. and E.A. Butcher, New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis. Journal of Sound and Vibration, (3): p [14] Mobarak, H.M. and H. WU, Crack breathing as a function of crack location under the effect of unbalance force. 24th Australasian Conference on the Mechanics of Structures and Materials (ACMSM24), Perth, Australia,

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