## 4 Two-Dimensional Finite Element Modelling and Vibration Analysis of Cracked Rotor  
The schematic in Figure 12 is representative of a cracked rotor model with two disks, supported by isotropic rolling-element bearings at both ends (nodes 1 and 21). The two-dimensional (2-D) finite element model is discretised into twenty Euler-Bernoulli beam elements (N = 20), where Element 17 is the cracked element and nodes 4 and 18 carry Disk 1 and Disk 2, respectively. An unbalanced masswith an unbalance eccentricity of \(1\times 10^{-5}\) kg\(\cdot\)m at an angle of \(0^{\circ}\) is attached to Disk 2. The physical specifications and bearing properties of the rotor are shown in Tables 3 and 4, respectively. The stiffness matrix of a homogenous Euler-Bernoulli beam element is written as  
\[[k_{p}]=\frac{E}{l^{3}}\left[ \begin{array}{cccccc}12I_{U}&0&0&6 Il_{U}&-12I_{U} &0&0&6 Il_{U}\\ 0&12I_{V}&-6 Il_{V}&0&0&-12I_{V}&-6 Il_{V}&0\\ 0&-6 Il_{V}&4I^{2}I_{V}&0&0&6 Il_{V}&2I^{2}I_{V}&0\\ 6 Il_{U}&0&0&4I^{2}I_{U}&-6 Il_{U}&0&0&2I^{2}I_{U}\\ -12I_{U}&0&0&-6 Il_{U}&12I_{U}&0&-6 Il_{U}\\ 0&-12I_{V}&6 Il_{V}&0&0&12I_{V}&6 Il_{V}&0\\ 0&-6 Il_{V}&2I^{2}I_{V}&0&0&6 Il_{V}&4I^{2}I_{V}&0\\ 6 Il_{U}&0&0&2I^{2}I_{U}&-6 Il_{U}&0&0&4I^{2}I_{U}\end{array} \right] \tag{19}\]  
where \(E\) is the elastic modulus of the shaft and \(l\) is the length of the cracked element [26]. The stiffness changes due to the breathing crack can be directly described by evaluating Equation (19) at chosen time intervals. Alternatively, applying a coordinate transformation to the \(\overline{X}\) and \(\overline{Y}\) axes allows the already derived \(I_{\overline{X}}\), \(I_{\overline{Y}}\) and \(I_{\overline{X}\overline{Y}}\) values to be used instead of the principle area moments of inertia. This is achieved by  
\[[k_{c}]=[T]\big{[}k_{p}\big{]}[T]^{T} \tag{20}\]  
where \([k_{c}]\) is the resulting cracked stiffness matrix about the \(\overline{X}\) and \(\overline{Y}\) axes and \([T]\) is the transformation matrix given by Equation (21), \begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Value** \\ \hline Bearing stiffness in x direction (N/m) & \(7\times 10^{7}\) \\ Bearing damping in x direction (Ns/m) & 500 \\ Bearing stiffness in y direction (N/m) & \(7\times 10^{7}\) \\ Bearing damping in y direction (Ns/m) & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Rolling-element bearing specifications of the rotor model. Figure 12: 2-D finite element model of a cracked rotor with disks at nodes 4 and 18. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Disk** & **Shaft** \\ \hline Material density (kg/m\({}^{3}\)) & 7800 & 7800 \\ Elastic modulus (GPa) & 210 & 210 \\ Thickness or length (m) & 0.015 & 1 \\ Outer radius (mm) & 63.5 & 12.7 \\ Inner radius (mm) & 12.7 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Physical specifications of the rotor model. \[[T]=\left[ \begin{array}{cccccccc}cos\!\left(\alpha_{p}\right)&-sin\!\left( \alpha_{p}\right)&0&0&0&0&0\\ sin\!\left(\alpha_{p}\right)&cos\!\left(\alpha_{p}\right)&0&0&0&0&0\\ 0&0&cos\!\left(\alpha_{p}\right)&-sin\!\left(\alpha_{p}\right)&0&0&0\\ 0&0&sin\!\left(\alpha_{p}\right)&cos\!\left(\alpha_{p}\right)&0&0&0\\ 0&0&0&0&cos\!\left(\alpha_{p}\right)&-sin\!\left(\alpha_{p}\right)&0&0\\ 0&0&0&0&sin\!\left(\alpha_{p}\right)&cos\!\left(\alpha_{p}\right)&0&0\\ 0&0&0&0&0&cos\!\left(\alpha_{p}\right)&-sin\!\left(\alpha_{p}\right)\\ 0&0&0&0&0&0&sin\!\left(\alpha_{p}\right)&cos\!\left(\alpha_{p}\right)\end{array}  \right]. \tag{21}\]  
The parameter \(\alpha_{p}\) is the rotation angle between the \(\overline{X}-\overline{Y}\) coordinate system and the principal centroidal coordinate system. The result of Equation (20) is equivalent to  
\[[k_{ce}]=\left[