# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 2 Theory
### Continuous Model Based on Rayleigh Model
Hence, the external force acting on the shaft by the disc and bearing is  
\[f_{y}=\sum_{i=1}^{n}f_{ib\cdot y}+\sum_{j=1}^{m}f_{id\cdot y} \tag{7}\]  
Figure 2.0 Multi-disc and multi-span rotor-bearing system. Figure 2.1 Physical model of the system. Figure 2.2  
Free-ended boundary condition can be used in the modeling. The flexural torque and shearing force at both ends are zero. Hence, Equations (9) and (10) can be obtained. \[\left\{ \begin{array}{rl}E\frac{\partial^{2}}{\partial x^{2}}y(z,t)|_{z=0,L}&= 0\\ E\frac{\partial^{3}}{\partial x^{2}}y(z,t)|_{z=0,L}&=0\\ \end{array} \right. \tag{9}\]  
\[\left\{ \begin{array}{rl}E\frac{\partial^{2}}{\partial x^{2}}x(z,t)|_{z=0,L}& =0\\ E\frac{\partial^{3}}{\partial x^{2}}x(z,t)|_{z=0,L}&=0\\ \end{array} \right. \tag{10}\]  
where \(L\) is the total length of rotor-shaft. To obtain dimensionless model, the following dimensionless quantities in Table 1 are defined. Therefore, the dimensionless partial differential equations of the lateral vibration of a rotating shaft motion in the unbalanced multi-disc and multi-span rotor-bearing system can be expressed in the form of Equations (11) and (12) according to Equations (1) and (2) and Equations (7) and (8). \[ \begin{array}{rl}\frac{\partial^{4}u}{\partial q^{4}}+\frac{mL^{4}}{EI} \frac{\partial^{2}u}{\partial l^{2}}-\left(\frac{I_{hp}L^{2}}{EI}\frac{ \partial^{4}u}{\partial q^{2}}\frac{\partial^{4}u}{\partial l^{2}}+\frac{I_{ hp}L^{2}w}{EI}\frac{\partial^{3}u}{\partial q^{2}}\right)=\frac{l^{3}}{EI}\left\{ \begin{array}{rl}\sum_{j=1}^{m}\left[\left(\pi m\eta_{ju}w^{2}e_{j}\sin(wt+ \alpha_{j})-m_{jd}L\frac{\partial^{2}u_{jd}}{\partial l^{2}}\right)\delta \left(L\left(q-q_{jd}\right)\right)\right]\\ -\sum_{i=1}^{n}\left[\left(k_{i\cdot xx}L\eta_{ib}+k_{i\cdot yy}Lu_{ib}+c_{i \cdot xx}L\frac{\partial u_{ib}}{\partial l}+c_{i\cdot yy}L\frac{\partial u_{ ib}}{\partial t}\right)\delta\left(L\left(q-q_{ib}\right)\right)\right] \end{array}\right\}\end{array}  \tag{11}\]  
\[ \begin{array}{rl}\frac{\partial^{4}v}{\partial q^{4}}+\frac{mL^{4}}{EI} \frac{\partial^{2}v}{\partial l^{2}}-\left(\frac{I_{hp}L^{2}}{EI}\frac{ \partial^{4}v}{\partial q^{2}\partial l^{2}}-\frac{I_{hp}L^{2}w}{EI}\frac{ \partial^{3}u}{\partial q^{2}\partial t}\right)=\frac{l^{3}}{EI}\left\{ \begin{array}{rl}\sum_{j=1}^{m}\left[\left(\pi m\eta_{ju}w^{2}e_{j}\cos(wt+ \alpha_{j})-m_{jd}L\frac{\partial^{2}u_{jd}}{\partial l^{2}}\right)\delta \left(L\left(q-q_{jd}\right)\right)\right]\\ -\sum_{i=1}^{n}\left[\left(k_{i\cdot xx}L\eta_{ib}+k_{i\cdot xy}Lu_{ib}+c_{i \cdot xx}L\frac{\partial u_{ib}}{\partial t}+c_{i\cdot yy}L\frac{\partial v_{ ib}}{\partial t}\right)\delta\left(L\left(q-q_{ib}\right)\right)\right] \end{array}\right\}\end{array}  \tag{12}\]  
where \(m_{jd}\) is the mass of #\(j\) disc; \(m_{ju}\) is the eccentric mass of #\(j\) disc; \(\alpha_{j}\) which is defined as the angle contained by the \(x\)-axis and the eccentric position in the direction of rotation is the eccentric angle of #\(j\) disc; \(e_{j}\) is #\(j\) disc's eccentric distance; \(\delta()\) is the Dirac's function; \(k_{i\cdot xx}\), \(k_{i\cdot xy}\), \(k_{i\cdot yy}\), \(k_{i\cdot yx}\) are the stiffness coefficients of #\(i\) bearing; \(c_{i\cdot xx}\), \(c_{i\cdot xy}\), \(c_{i\cdot yy}\), \(c_{i\cdot yx}\) are the damping coefficients of #\(i\) bearing.