## 4 Two-Dimensional Finite Element Modelling and Vibration Analysis of Cracked Rotor
These crack depths were chosen as \(\mu\) = 0 means no crack is present, \(\mu\) = 0.2 is an early/mid stage crack, \(\mu\) = 0.5 is a mid-stage crack and \(\mu\) = 0.8 is a late stage crack. The notation 'nB' and 'nF' refers to the backwards and forwards damped natural frequencies for order n, respectively; for example, 2B is the second backwards damped natural frequency and 3F is third forwards damped natural frequency. It is explained in literature [29; 30] that backwards whirl modes result in a lower damped natural frequency relative to forward whirl modes due to the gyroscopic effect of the rotor disks and therefore the two curves will diverge from one another. Additionally, Figure 13b-d shows that for cracked rotors each corresponding forwards and backwards whirl frequency lines are separated at zero RPM, whereas for an undamaged shaft (Figure 13a), the frequency value at zero RPM is identical. Furthermore, the asymmetry of the open crack cross section and corresponding stiffness asymmetries causes the backwards and forwards natural frequency pairs to further separate. Figure 13 also features a diagonally spanning dotted line representative of the first-order excitation. The intersection points between the excitation line and the damped natural frequency lines determine the set of forward and backward critical speeds of each rotor. Table 5 summarises the critical speed values found according to Figure 13.  
Figures 14 and 15 show the vertical amplitude (Y-direction) of Node 1 for the first three harmonic frequency components predicted by the proposed model and the chosen HNA model. Rotor spin speeds within the vicinity of the first and second critical speeds were chosen, as well as crack depths of \(\mu\) = 0, 0.2, 0.5 and 0.8. The series of peaks in frequency amplitude that appear throughout the figures approximately correspond to discrete fractions of the critical whirl speeds shown in Table 5, for example, \({}^{1}/_{3}\omega_{2}\) is one-third of the second forwards critical speed. These peaks occur in pairs where the left and right peaks correspond to a discrete fraction of a particular backwards and forwards critical speed, respectively. At times, these peaks appear to be singular due to the closeness of the peaks and inadequate granularity of the horizontal-axis and so are labelled without 'b' and 'f', for example \({}^{\Lambda}/_{3}\omega_{2}\); instead of \({}^{\Lambda}/_{3}\omega_{2}\); and \({}^{\Lambda}/_{3}\omega_{\rm b2}\). Moreover, the isolated peaks may be due to the pair of peaks being greatly different in value. For an undamaged shaft (\(\mu\) = 0), both models show that a peak occurs at the first and second critical speeds, \(\omega_{1}\) and \(\omega_{2}\), for the 1X, 2X and 3X frequencies. Furthermore, for the 2X and 3X frequencies, peaks occur at one-half and one-third the first three critical speeds, respectively. The occurrence of these peaks is therefore unrelated to fatigue cracks; however, it is known in the literature that the magnitude of these peaks may increase depending on unbalance [31] and crack depth [32; 33]. For crack depths of \(\mu\) = 0.2 and 0.5, peaks relating to other discrete fractions of critical speeds arise in addition to the aforementioned peaks. As the crack deepens, the magnitude of these additional peaks increases, suggesting that these peaks are related to the presence of fatigue cracks. In fact,