## 4 Two-Dimensional Finite Element Modelling and Vibration Analysis of Cracked Rotor
\begin{array}{cccccccc}12I_{11}&0&0&6I_{11}&-12I_{1 1}&0&0&6I_{11}\\ 0&-12I_{11}&6I_{11}&0&0&12I_{11}&6I_{11}&0\\ 0&6I_{11}&-4I^{2}I_{11}&0&0&-6I_{11}&-2I^{2}I_{11}&0\\ 6I_{11}&0&0&4I^{2}I_{11}&-6I_{11}&0&0&2I^{2}I_{11}\\ -12I_{11}&0&0&-6I_{11}&12I_{11}&0&0&-6I_{11}\\ 0&12I_{11}&-6I_{11}&0&0&-12I_{11}&-6I_{11}&0\\ 0&6I_{11}&-2I^{2}I_{11}&0&0&-6I_{11}&-4I^{2}I_{11}&0\\ 6I_{11}&0&0&2I^{2}I_{11}&-6I_{11}&0&0&4I^{2}I_{11}\end{array} \right], \tag{27}\]  
\[[k_{2}]=\frac{E}{\beta}\left[ \begin{array}{cccccccc}0&0&0&0&0&0&0\\ 0&12I_{22}&-6I_{22}&0&0&-12I_{22}&-6I_{22}&0\\ 0&-6I_{22}&4I^{2}I_{22}&0&0&6I_{22}&2I^{2}I_{22}&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&-12I_{22}&6I_{22}&0&0&12I_{22}&6I_{22}&0\\ 0&-6I_{22}&2I^{2}I_{22}&0&0&6I_{22}&4I^{2}I_{22}&0\\ 0&0&0&0&0&0&0&0\end{array} \right], \tag{28}\]  
\[[k_{3}]=\frac{E}{\beta}\left[ \begin{array}{cccccccc}0&-12I_{33}&6I_{33}&0&0 &12I_{33}&6I_{33}&0\\ -12I_{33}&0&0&-6I_{33}&12I_{33}&0&0&-6I_{33}\\ 6I_{33}&0&0&4I^{2}I_{33}&-6I_{33}&0&0&2I^{2}I_{33}\\ 0&-6I_{33}&4I^{2}I_{33}&0&0&6I_{33}&2I^{2}I_{33}&0\\ 0&12I_{33}&-6I_{33}&0&0&-12I_{33}&-6I_{33}&0\\ 12I_{33}&0&0&6I_{33}&-12I_{33}&0&0&6I_{33}\\ 6I_{33}&0&0&2I^{2}I_{33}&-6I_{33}&0&0&4I^{2}I_{33}\\ 0&-6I_{33}&2I^{2}I_{33}&0&0&6I_{33}&4I^{2}I_{33}&0\end{array} \right] \tag{29}\]  
where \(I_{11}=I-\bar{I}_{1}\), \(I_{22}=2I-\bar{I}_{1}-\bar{I}_{2}\) and \(I_{33}=(\bar{I}_{2}-\bar{I}_{1})/2\). It should be noted the final expression of \([k_{\sigma}]\) is independent of \(\alpha_{p}\) so the angle does not need to be determined. The stiffness element matrix for the non-cracked elements, as well as the mass and damping/gyroscopic matrices for all the shaft elements, disk and rolling-element bearings can be found in literature [29]. The equation of motion associated with the cracked rotor system is described by  
\[Mq^{\prime\prime}(t)+(C+\Omega G)q^{\prime}(t)+(K_{0}+K_{1}f_{1}(t)+K_{2}f_{2} (t)+K_{3}f_{3}(t))q(t)=F_{1}cos(\Omega t)+F_{2}sin(\Omega t)+F_{g} \tag{30}\]  
where \(M\), \(C\), \(G\), \(K_{0}\), \(K_{1}\), \(K_{2}\) and \(K_{3}\) are global matrices of size 4 (N + 1) \(\times\) 4 (N + 1) relating to the mass, damping, gyroscopic effect, \([k_{0}]\), \([k_{1}]\), \([k_{2}]\) and \([k_{3}]\), respectively. The global matrices are assembled using a standard finite element modelling procedure [29]. The assembly of \(K_{0}\) includes the stiffness of all the shaft elements; however, stiffness element entries associated with the cracked element are replaced by those in \([k_{0}]\). Additionally, \(K_{1}\), \(K_{2}\) and \(K_{3}\), are sparse matrices whose only non-zero elements correspond to the values of \([k_{1}]\), \([k_{2}]\) and \([k_{3}]\), respectively. The 4 (N + 1) \(\times\) 1 displacement vector \(q(t)\) describes the nodal translations and nodal rotations \(\left[x_{i}\,y_{i}\,\phi_{i}^{x}\,\phi_{i}^{y}\right]\) about the \(X\) and \(Y\) axes. The 4 (N + 1) \(\times\) 1 vector \(F_{1}\) and \(F_{2}\) are the unbalance amplitude vectors and \(F_{g}\) is the weight force vector [11]. The equations of motion of the cracked rotor system can be solved by using a harmonic balance procedure by assuming that the solution of \(q(t)\) is in the following form:  
\[q(t)=A_{0}+\sum_{k=1}^{n}(A_{k}cos(k\Omega t)+B_{k}sin(k\Omega t)) \tag{31}\]  
where \(\overline{Q}\) is the first-order system corresponding to Equation (30), \(\overline{Z}\) is an 4 (N + 1) \(\times\) 4(N + 1) matrix of zeroes and \(I\) is the 4 (N + 1) \(\times\) 4 (N + 1) identity matrix. Since the stiffness \(K_{0}\) is used, the eigenvalue solutions of \(\overline{Q}\) provides the damped natural frequency of a rotor with a non-breathing open crack. The Campbell diagrams in Figure 13 illustrate the change in the first four forwards and backwards damped natural frequencies (given in Hz) with an increase in rotor spin speed (given in rev/min) for open cracks of different depths, namely \(\mu\) = 0, 0.2, 0.5 and 0.8.