# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 3 Numerical Simulations and Discussion
### Calculated Unbalance Response
#### 3.3.2 Discussion
Figure 18: _Cont._  
In addition, for Ricatti method, the rotor is simplified as a beam mode with lumped mass, which causes errors in modelling and the inaccuracy of calculating unbalance response. While for FEM, the model is closer to real rotors and unbalance responses obtained by FEM are more accuracy than Ricatti method because the factors such as moment of inertia, gyroscopic moment and shear deformation are considered. CRDAM is closer to FEM than Ricatti. Hence, CRDAM can be considered as an effective method for calculating unbalance responses of rotors. According to the above conclusion, the unbalance responses of all points of g1.5 and g5.5 at the frequencies which are away from the critical frequencies, are obtained in Figures 19 and 20. For h1.5, the frequencies 1 Hz, 10 Hz, 25 Hz, 45 Hz, 65 Hz, 100 Hz, 130 Hz, 220 Hz, 180 Hz and 280 Hz are selected. For h5.5, they are 100 Hz, 300 Hz and 400 Hz. According to Figure 19, it is found that using CRDAM, FEM and Ricatti, the three obtained curves, which represent the relation between unbalance responses and positions of the 61 points on the shaft at a fixed rotating frequency, are well coincident. The curve obtained by Ricatti are well coincident with the CRDAM and FEM curves when the frequency is 1 Hz, 10 Hz, 25 Hz, 45 Hz, 65 Hz, 100 Hz, 130 Hz and 220 Hz, but not well coincident with the two curves at 180 Hz and 280 Hz. According to Figure 19h, the curve obtained by Ricatti are not coincident with the other two curves at #48-49 points when  
by Ricatti are not coincident with the other two curves at #11-12 points in Figure 20c. It indicates that the CRDAM method is closer to FEM than Ricatti. 2  
Numerical stability  
According to simulation of g4.4, unbalance response angle obtained by CRDAM changes frequently and suddenly from positive to negative with frequency when the frequency is high. It indicates that the numerical instability occurs and the unbalance response cannot be correctly calculated by the proposed method. Moreover, similar phenomena can be found in simulations of g5.5, g1.4, g1.5, h4.4, h5.5, h1.4 and h1.5 whose results are shown in the Figures 21-27. According to Figure 21 obtained using g5.5, the phenomenon of numerical instability of CRDAM is obviously when the rotating frequency is higher than 800 Hz. According to Figure 22 obtained using g1.4, numerical instability of CRDAM occurs after 600 Hz and becomes more serious when the frequency increases. For g1.5, numerical instability of CRDAM occurs after 400 Hz and becomes more serious when the frequency increases according to Figure 23. For h4.4, the numerical instability occurs at 630 Hz and becomes obvious from 700 Hz to 1000 Hz according to Figure 24. According to Figure 25, numerical instability of CRDAM in h5.5 occurs after 500 Hz. According to Figure 26, it occurs after 580 Hz and becomes more serious from 650 Hz to 1000 Hz. According to Figure 27, it occurs after 400 Hz and becomes more serious from 500 Hz to 1000 Hz. Whereas for FEM and Ricatti, there is no numerical instability phenomenon although their curves still do not coincide according to Figures 16 and 21-27. Figure 24: Calculated unbalance response of point #2 in h4.4: (**a**) 600–700 Hz; (**b**) 700–1000 Hz. Figure 23: Calculated unbalance response of point #2 in g1.5: (**a**) 320–500 Hz; (**b**) 500–1000 Hz. Figure 22: Calculated unbalance response of point #2 in g1.4: (**a**) 500–750 Hz; (**b**) 750–1000 Hz. When Matlab performs inversion of the matrix \(M_{11}\) at some frequencies, the alarm that results may be inaccurate because \(M_{11}\) is close to singular or badly scaled occurs. Therefore, the reciprocal condition of a matrix is used to investigate the numerical stability which is caused by the singular matrix \(M_{11}\). If a matrix is well conditioned, its reciprocal condition number is near 1. If a matrix is badly conditioned, its reciprocal condition is near 0. Figures 28-31 which show the reciprocal condition numbers of \(M_{11}\), are obtained by the function round( ) of Matlab.