# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 3 Numerical Simulations and Discussion  
### Methodology of Numerical Simulations  
The aim of the numerical simulation is to validate the proposed CRDAM. The Ricatti and FEM results are taken as the references. The Ricatti method, which has the advantage of fast calculation speed and low requirement for CPU, is an old classical method to obtain rotor dynamic characteristics. With the rapid development of computes, the FEM method becomes an effective and commonly used method for calculating rotor dynamic characteristics of rotors, especially for large rotors. In the simulations, critical frequencies and unbalance responses of the three methods are obtained to conduct the comparison among them for the validation. When relative errors and absolute errors are used to quantitatively analyze the difference of calculated critical frequencies and unbalance responses among three method, FEM results are treated as the base-exact.  
18 computational examples are used in the simulation. Figures 4-12 shows nine kinds of rotors. They represent five-span and five-disc rotor, single-span and five-disc rotor, four-span and four-disc rotor, three-span and three-disc rotor, two-span and two-disc rotor, single-span and single-disc rotor, single-span and four-disc rotor, single-span and three-disc rotor and single-span and two-disc rotor. The 18 computational examples are the nine kinds rotors with oil-journal bearings and nine kinds rotors with rolling bearings. g1.1, g1.2, g1.3, g1.4, g1.5, g2.2, g3.3, g4.4 and g5.5 are the nine kinds of rotors with rolling bearings and h1.1, h1.2, h1.3, h1.4, h1.5, h2.2, h3.3, h4.4 and h5.5 the nine kinds rotors with oil-journal bearings. The parameters of the bearings, the disc and shaft are listed in Tables 2-6.  
Figure 8: Two-span and two-disc rotor.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Meaning** \\ \hline r\_shaft & Radius of the shaft \\ p\_shaft & Density of the shaft \\ E\_shaft & Elastic modulus of the shaft \\ L\_shaft & Length of the shaft \\ r\_disc & Radius of the disc \\ p\_disc & Density of the disc \\ E\_disc & Elastic modulus of the disc \\ L\_disc & Width of the disc \\ \hline \hline \end{tabular}
\end{table}
Table 2: Meanings of symbols in the computational example of the rotor.  
Figure 10: Single-span and four-disc rotor.  
Figure 9: Single-span and two-disc rotor.  
Figure 12: Single-span and two-disc rotor.  
Based on the CRDM, FEM [10] and Ricatti [11] methods, a calculation program to calculate the unbalance responses is developed using Matlab software. As shown in Figures 4-12, the rotor-shaft is divided into 60 unit elements of equal length and the discs and bearings are lumped when unbalance responses of FEM and Ricatti methods are  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{Z_{1b}}\) & \(\mathbf{Z_{2b}}\) & \(\mathbf{Z_{3b}}\) & \(\mathbf{Z_{4b}}\) & \(\mathbf{Z_{5b}}\) &  
calculated by the program. The #2 (2 th) point's unbalance responses changed with the rotating frequencies (speeds) are obtained for the comparison among the three methods. In the computation, the calculating frequency is from 0 to 1000 Hz and the interval is 1 Hz.