# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 4 Conclusions  
For the multi-disc and multi-span rotor-bearing system, CRDAM, which can be used for further studies on identification of rotor unbalance and bearing coefficients, is proposed. Numerical experiments are conducted to validate the proposed method by comparing it with FEM and Ricatti. The results of this paper are concluded as follows:  
1. The functional relationship between the unbalance response and location on the shaft, rotor unbalance (amplitude and angle), each bearing's stiffness and damping coefficients and rotor's inherent parameters is obtained based on CRDAM. 2. Numerical simulations indicate that the obtained unbalance response calculated by the three methods are almost equal when the frequency is away from the critical frequency, but they are different when the frequency is near the critical frequency. Moreover, CRDAM is closer to FEM than Ricatti. Although the critical frequencies calculated by them are almost equal to each other, the critical frequency calculated by CRDAM is closer to the critical frequency calculated by FEM than the critical  
Figure 32: Time consumption of calculating the unbalance response from 0 Hz to 1000 Hz by CRDAM, FEM and Ricatti methods using the 18 computation examples: (**a**) for rotors with rolling bearings; (**b**) for rotors with oil journal bearings. frequency obtained by Ricatti. The unbalance response obtained by CRDAM is also closer to the unbalance response obtained by FEM than the unbalance response by Ricatti, especially when the frequency is smaller than the low order frequency. 3. Numerical simulations indicate that the calculating speed of CRDAM is the second fast among the three methods, but the speed of CRDAM becomes faster than Ricatti when the rotor is complicated. Simulations also show that the numerical stability of FEM and Ricatti is better than CRDAM. However, the numerical instability of CRDAM occurs when the frequency is high, often higher than the fifth order frequency. In engineering practice, actual rotors work at the speed, which must be away from each critical frequency, to avoid sympathetic vibration. Hence, CRDAM is applicable to calculating unbalance response of multi-disc and multi-span rotor. High speed rotors are just across third critical frequency. Therefore, the proposed method can be applied to actual high speed rotors. However, complex rotor systems, such as the reducer gear transmission systems, which have multi shafts parallel in space, cannot be modelled by the proposed method. FEM and Ricatti should be applied for these complex rotors. For further study, the loss factor could be considered for modelling the continuous rotor. Timoshenko model could be used in the model of the rotor. The forward problems of the continuous rotor dynamics for the multi-disc and multi-span rotors, is solved in this paper. Inverse problems of the continuous rotor dynamics could be conducted based the proposed CRDAM. For example, identification of rotor unbalance and bearing stiffness and damping of without test runs and external excitations, which has the advantages of easy implementation and low cost, should be studied in the future. It can be achieved by solving equations which are built using unbalance responses at some positions on rotor-shaft as known conditions based on the proposed functional relationship. Conceptualization, A.W. ; methodology, A.W. ; software, A.W., Y.B. and Y.X. ; formal analysis, A.W. ; investigation, Y.X., X.C. and J.Y. ; writing--original draft preparation, A.W. and Y.B. ; writing--review and editing, A.W. and Y.B. ; supervision, G.M. ; funding acquisition, G.M. All authors have read and agreed to the published version of the manuscript. This research was funded by the Fundamental Research Funds for the Central Universities, grant number 8000150A084 and the Yue Qi Young Scholar Project, China University of Mining and Technology, Beijing, grant number 800015Z1145. Not applicable. The study did not report any data. The authors declare no conflict of interest. \[M_{12}=\left[