## 2 Modelling and Validation of a Two-Disc Rotor with a Breathing Crack
The first three natural frequencies are obtained and listed in Table 2. It should be noted that the results from the beam model and ABAQUS software were acquired after a few rounds of model updating iterations by adjusting bearing parameters to make the errors of the first three mode frequencies between ABAQUS and measured ones as small as possible. The corresponding bearing stiffness was found to be 960,000 Nm\({}^{-1}\), and bearing damping is 100 Nsm\({}^{-1}\). As one can see from Table 2, the maximum error between the beam model and ABAQUS was 4.17%, and the error between the beam model and measured results was even less than 2%. The above results show that the established beam model is validated and can be used for further dynamic research. It will be used to study the dynamic behaviors of rotors with cracks, provided that the crack model is accurate. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature** & **Value (Units)** & **Feature** & **Value (Units)** \\ \hline Shaft length (\(l_{0}\)) & 0.6 m & Mass of disc 1 & 0.759 kg \\ \(l_{1}\) & 0.46 m & polar moment of inertia of disc 1 & \(5.758\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ \(l_{2}\) & 0.07 m & diametrical moment of inertia of disc 1 & \(3.177\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ \(l_{3}\) & 0.08 m & Mass of disc 2 & 0.770 kg \\ \(l_{4}\) & 0.1 m & polar moment of inertia of disc 2 & \(5.843\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ Shaft diameter (\(D\)) & 0.01 m & diametrical moment of inertia of disc 2 & \(3.232\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ density (40 Cr) & \(7.87\times 10^{3}\) kg\(\cdot\)m\({}^{-3}\) & Gravitational acceleration & 9.8 m/s\({}^{2}\) \\ Yang’s modulus & \(2.11\times 10^{11}\) Pa & Rayleigh damping coefficient (\(a\)) & 0.68 \\ Poisson’s ratio & 0.277 & Rayleigh damping coefficient (\(b\)) & \(2.8\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the two-disc rotor. Based on the validated model, the Newmark integration method [57] was adopted to compute the rotor unbalance response after assembling a crack into the rotor by the afore-mentioned crack breathing model. To illustrate the dynamic behavior of the cracked two-disc rotor, a typical waterfall diagram of the cracked rotor (with a crack depth of 0.35\(D\), located in the 30 th element) under different rotating speeds below the first critical speed of the rotor is shown in Figure 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Order** & \begin{tabular}{c} **Beam** \\ **Model** \\ \end{tabular} & **ABAQUS** & **Measured** & \
\begin{tabular}{c} **Error between** \\ **Beam Model** \\ **and ABAQUS** \\ \end{tabular} &
\begin{tabular}{c} **Error between** \\ **Beam Model and** \\ **Measured Results** \\ \end{tabular} \\ \hline
1 st vertical bending & 42.1 Hz & 40.9 Hz & 42.5 Hz & 1.2 Hz (2.85\%) & 0.4 Hz (0.95\%) \\
1 st lateral bending & 42.1 Hz & 41.0 Hz & 42.9 Hz & 1.1 Hz (2.61\%) & 0.8 Hz (1.90\%) \\
2 nd vertical bending & 107.9 Hz & 112.4 Hz & 107.4 Hz & 4.5 Hz (4.17\%) & 0.5 Hz (0.47\%) \\
2 nd lateral bending & 107.9 Hz & 112.4 Hz & 108.7 Hz & 4.5 Hz (4.17\%) & 0.8 Hz (0.74\%) \\
1 st torsion & 158.8 Hz & 160.2 Hz & - & 1.4 Hz (0.88\%) & - \\
3 rd vertical bending & 349.6 Hz & 358.0 Hz & 349.6 Hz & 8.4 Hz (2.40\%) & 0.0 Hz (0.00\%) \\
3 rd lateral bending & 349.6 Hz & 357.2 Hz & 350.7 Hz & 7.6 Hz (2.17\%) & 1.1 Hz (0.31\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the natural frequencies from numerical calculation and experiment. Figure 3: Waterfall diagram of the two-disc rotor (**a**) rotating speed near 1/3 critical speed; (**b**) rotating speed near 1/2 critical speed. Figure 2: Rotor-bearing system for modal experiment. (**1**) Motor; (**2**) flexible coupling; (**3**) disc; (**4**) eddy current displacement sensors; (**5**) bearings. As one can see from Figure 3, \(2\times\), and \(3\times\) super-harmonic components appear and do not exist in an un-cracked rotor, because of crack breathing, which is a nonlinear feature.