## 4 Two-Dimensional Finite Element Modelling and Vibration Analysis of Cracked Rotor
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mu\)} & \(\mathbf{\omega_{\rm{b1}}}\) & \(\mathbf{\omega_{\rm{f1}}}\) & \(\mathbf{\omega_{\rm{b2}}}\) & \(\mathbf{\omega_{\rm{f2}}}\) & \(\mathbf{\omega_{\rm{b3}}}\) & \(\mathbf{\omega_{\rm{f3}}}\) & \(\mathbf{\omega_{\rm{b4}}}\) & \(\mathbf{\omega_{\rm{f4}}}\) \\  & **(rpm)** & **(rpm)** & **(rpm)** & **(rpm)** & **(rpm)** & **(rpm)** & **(rpm)** & **(rpm)** & **(rpm)** \\ \hline
0 & 2616 & 2666 & 8416 & 8594 & 18,443 & 18,577 & 34,042 & 38,238 \\
0.2 & 2610 & 2662 & 8358 & 8554 & 18,290 & 18,516 & 34,018 & 38,102 \\
0.5 & 2582 & 2646 & 8072 & 8465 & 17,686 & 18,374 & 33,926 & 37,644 \\
0.8 & 2504 & 2624 & 7446 & 8300 & 16,730 & 18,055 & 33,726 & 36,918 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: First four forwards and backwards critical whirl speeds. Figures 14 and 15 show that there is an overall increase in the magnitudes of the 2X and 3X components for \(\mu\) = 0.2, 0.5 and 0.8 relative to no crack. An upwards trending increase to the 2X and 3X components is well known in the literature as a theoretical and practical method for determining the presence of fatigue cracks [2]. With regards to magnitude, the 1X, 2X and 3X frequency amplitudes predicted by the two models are identical or approximately equal for crack depths of \(\mu\) = 0, 0.2 and 0.5. However, at particular rotor speeds, the two models significantly differ in magnitude, such as at points of interest A, B, C and D seen in Figures 14b and 15b,c, respectively. Since numerical instability can lead to misleading results, the numerical stability of the rotor dynamic solutions at the points of interest was examined. The numerical stability of a cracked rotor can be found quickly in the frequency domain by identifying the sign of the determinant after scaling the coefficient matrix \(\delta\) in Equation (35), as shown in [34]. Figure 16 displays the transverse trajectory of Node 1 at the four points of interest and also indicates the numerical stability of the solution, or lack thereof. It was found that at point of interest A (2800 rpm, \(\mu\) = 0.2), the HNA model's solution was numerically unstable; however, the proposed model's solution was stable. Conversely, the proposed model was unstable and the HNA model was stable at point C (7255 rpm, \(\mu\) = 0.5). Visual inspection of points A and C in Figures 14 and 15 show that these peaks rise sharply and suddenly occur (as opposed to gradually forming) and the other model does not predict a sudden peak at the corresponding rotor speed. Therefore, it is suggested that these peaks occur due to numerical instability and thus explains  
Figure 13: Campbell diagrams for cracked rotor: (**a**) \(\mu\) = 0, (**b**) \(\mu\) = 0.2, (**c**) \(\mu\) = 0.5 and (**d**) \(\mu\) = 0.8.  
the difference between the two models at these points of interest. On a side note, the similarities between the two models for early/mid staged cracks (\(\mu\) = 0.2 and 0.5) may be exploited to determine a numerically stable vibration solution when only one model is found to produce an unstable result. In contrast, both the proposed model and the HNA model had stable solutions for points of interest B (8455 rpm, \(\mu\) = 0.2) and D (8235 rpm, \(\mu\) = 0.5), suggesting that the small difference in the breathing mechanics of the two crack models is sufficient to result in relatively different vibration behaviour for early/mid stage cracks (\(\mu\) = 0.2 and 0.5). Furthermore, the transverse trajectory of Node 1 at points of interest B and D also differed in shape; therefore, these points of interest may be a meaningful indicator when experimentally validating the proposed model. Figures 14d and 15d show the 1X, 2X and 3X frequency amplitudes for the two models at a crack of depth \(\mu\) = 0.8. Across all three frequencies, both models have no clear global maximum amplitude value due to the appearance of a high number of amplitude peaks and also both models have very few rotor speeds for which a common peak occurs.