# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 2 Theory  
### Continuous Model Based on Rayleigh Model  
The multi-disc and multi-span rotor-bearing system, which has \(m\) discs and \(n\) bearings, is shown in Figure 1. The force analysis of micro-element is shown in Figure 2. Regarding the connection between shafts as a rigid connection and using [PERSON]'s theory [14; 15; 16; 33; 43] in which the shaft is regarded a perfect elastic body (It is homogeneous and isotropic and obeys Hooke's Law) and the moment of inertia of rotor-shaft is considered, the relationship between the lateral displacement of rotor-shaft in \(x\) and \(y\) directions and the excitation can be described as following according to Figure 2.  
\[EI\frac{\partial^{4}y}{\partial z^{4}}+m\frac{\partial^{2}y}{\partial t^{2}}- \left(I_{dx}\frac{\partial^{4}y}{\partial z^{2}\partial t^{2}}+I_{p}w\frac{ \partial^{3}x}{\partial z^{2}\partial t}\right)=f_{y} \tag{1}\]  
\[EI\frac{\partial^{4}x}{\partial z^{4}}+m\frac{\partial^{2}x}{\partial t^{2}}- \left(I_{dy}\frac{\partial^{4}x}{\partial z^{2}\partial t^{2}}-I_{p}w\frac{ \partial^{3}y}{\partial z^{2}\partial t}\right)=f_{x} \tag{2}\]  
where \(x\) and \(y\) are the lateral displacement components due to lateral vibrations of the rotor shaft in the natural coordinates; \(z\) is the axial position of the shaft; \(t\) is the time; \(m\) is the mass per unit length of rotor-shaft; \(w\) is the rotation frequency; \(I_{p}\) is the polar moment of inertia per unit length of the shaft; \(I_{dx}\) is the diameter moments of inertia per unit length of the shaft in the \(x\) directions. \(\theta_{x}(z,t)\) and \(\theta_{y}(z,t)\) is the rotation angle of the micro-element around \(x\)-axis and \(y\)-axis, respectively; \(T_{y}\) is the shear force on the left end of the micro-element in \(y\) direction; \(M_{x}\) is the moment on the left end of the micro-element in \(y\)oz plane; \(f_{y}\) is the external force in \(y\) direction on the micro-element of the shaft; \(I_{dy}\) is the diameter moments of inertia per unit length of the shaft in the \(y\) directions; \(T_{x}\) is the shear force on the left end of the micro-element in \(x\) direction; \(M_{y}\) is the moment on the left end of the micro-element in \(x\)oz plane; \(f_{x}\) is the external force in \(x\) direction on the micro-element of the shaft; \(E\) is the elastic modulus of the shaft; \(I\) is the diametric shaft cross-sectional geometric moment of inertia. The force of each bearing acting on the shaft can be expressed as following. \[f_{ib\cdot y}=\bigg{(}k_{i\cdot xx}x_{ib}+k_{i\cdot yy}y_{ib}+c_{i\cdot yx}\frac{ \partial x_{ib}}{\partial t}+c_{i\cdot yy}\frac{\partial y_{ib}}{\partial t} \bigg{)}\delta(z-z_{ib}) \tag{3}\]  
\[f_{ib\cdot x}=\bigg{(}k_{i\cdot xx}x_{ib}+k_{i\cdot xy}y_{ib}+c_{i\cdot xx} \frac{\partial x_{ib}}{\partial t}+c_{i\cdot xy}\frac{\partial y_{ib}}{\partial t }\bigg{)}\delta(z-z_{ib}) \tag{4}\]  
where \(i\) is from 1 to \(n\); \(k_{i\cdot xx}\), \(k_{i\cdot xy}\), \(k_{i\cdot yx}\) are the four stiffness coefficients of #i bearing; \(c_{i\cdot xx}\), \(c_{i\cdot xy}\), \(c_{i\cdot yy}\) and \(c_{i\cdot yx}\) are the four damping coefficients of #i bearing; \(z_{ib}\) is the \(z\) coordinate position of #i bearing; \(y_{ib}\) and \(x_{ib}\) are the lateral displacement components due to lateral vibrations of the rotor shaft in position \(z_{ib}\). The force of the eccentric disc acting on the shaft can be expressed as following. \[f_{id\cdot y}=\bigg{(}m_{ju}w^{2}e_{j}\sin\big{(}wt+\alpha_{j}\big{)}-m_{j \cdot d}\frac{\partial^{2}y_{id}}{\partial t^{2}}\bigg{)}\delta\Big{(}z-z_{ jd}\Big{)} \tag{5}\]  
\[f_{id\cdot x}=\bigg{(}m_{ju}w^{2}e_{j}\cos\big{(}wt+\alpha_{j}\big{)}-m_{j \cdot d}\frac{\partial^{2}x_{jd}}{\partial t^{2}}\bigg{)}\delta\Big{(}z-z_{ jd}\Big{)} \tag{6}\]  
where \(j\) is from 1 to \(m\); \(m_{id}\) is the mass of #j disc; \(m_{ju}\) is the eccentric mass of #j disc; \(e_{j}\) is the eccentric distance of #j disc; \(\alpha_{j}\) is the eccentric angle of #j disc; \(z_{jd}\) is the \(z\) coordinate position of #j disc.