## 3 Algorithm Considering Centrifugal Force and Temperature Gradient  
### Centrifugal Force  
Multi-layer interference cylinder is often used to transmit torque. Under high speed, the influence of centrifugal force on contact pressure of multi-layer interference cylinder cannot be ignored.  
According to elastic mechanics, taking the small unit as a micro unit in polar coordinate radial plane of arbitrary radius \(\rho\), its force diagram is shown in Figure 3.  
Figure 3: A schematic diagram of the centrifugal force of a micro unit.  
Figure 2: The force diagram of interference cylinder.  
It is assumed that the density of micro unit is \(\rho_{0}\), the centrifugal force of the micro unit is:  
\[K_{\rho}=\rho\omega^{2}\rho_{0} \tag{3}\]  
The equilibrium differential equation of the micro unit is [18]:  
\[\frac{d\sigma_{\rho}}{d\rho}+\frac{\sigma_{\rho}-\sigma_{\varphi}}{\rho}+\rho \omega^{2}\rho_{0}=0 \tag{4}\]  
According to the geometric equation, radial strain \(\varepsilon_{\rho}\) and circumferential strain \(\varepsilon_{\varphi}\) are given as:  
\[\varepsilon_{\rho}=\frac{du}{d\rho},\varepsilon_{\varphi}=\frac{u}{\rho} \tag{5}\]  
Radial stress and circumferential stress of the micro unit can be obtained:  
\[\begin{array}{l}\sigma_{\rho}=\frac{E}{1-\mu^{2}}\big{(}\varepsilon_{\rho}+ \mu\varepsilon_{\varphi}\big{)}\\ \sigma_{\varphi}=\frac{E}{1-\mu^{2}}\big{(}\varepsilon_{\varphi}+\mu \varepsilon_{\rho}\big{)}\end{array} \tag{6}\]  
The equilibrium differential equation expressed by displacement can be obtained according to Equations (4)-(6).  
\[\frac{d^{2}u}{d\rho}+\frac{1}{\rho}\frac{du}{d\rho}-\frac{u}{\rho^{2}}=-\frac{ 1-\mu^{2}}{E}\rho\omega^{2}\rho_{0} \tag{7}\]  
The Equation of the displacement of a free cylinder at any point under centrifugal force can be obtained by solving Equation (7).  
\[u_{s}=\frac{(3+\mu)\rho_{0}\omega^{2}}{8E\rho}\Big{[}(1-\mu)\Big{(}a^{2}+b^{2 }\Big{)}\rho^{2}+(1+\mu)a^{2}b^{2}\Big{]}-\frac{\rho_{0}\big{(}1-\mu^{2}\big{)} \omega^{2}\rho^{3}}{8E} \tag{8}\]  
It is assumed that \(L_{n,i}\) is radial displacement caused by centrifugal force, \(n\)= 1 or 2, \(i\)= 1, 2, 3,...; \(n=1\) means inner surface, \(n=2\) means outer surface; \(i\) means cylinder number.  
Transforming Equation (8) into diameter direction, the radial displacement of outer surface of cylinder \(C_{i}\) based on centrifugal force is shown as:  
\[\begin{array}{ll}L_{2,i}&=\bigg{\{}\frac{(3+\mu_{i})\rho_{0}}{32E_{i}4\rho_ {2i}}\bigg{[}(1-\mu_{i})\big{(}{d_{1,i}}^{2}+{d_{2,i}}^{2}\big{)}{d_{2,i}}^{2 }+(1+\mu_{i}){d_{1,i}}^{2}{d_{2,i}}^{2}-\frac{\rho_{0}\big{(}1-\mu_{i}^{2} \big{)}{d_{2,i}}^{3}}{32E_{i}}\bigg{]}\bigg{\}}\omega^{2}\\ &=G_{2,i}\omega^{2}\end{array} \tag{9}\]  
Similarly, radial displacement of the inner surface of cylinder \(C_{i+1}\) based on centrifugal force is shown as:  
\[L_{1,i+1}=G_{1,i+1}\omega^{2} \tag{10}\]  
According to Equations (2), (9), and (10), the matrix model of interference magnitude of multi-layer interference cylinder under the centrifugal force can be obtained:  
\[\textbf{ffi}=\textbf{\emph{K}}\textbf{\emph{p}}+\textbf{\emph{G}}\omega^{2} \tag{11}\]  
Among them, \(\textbf{\emph{G}}=[G_{1,2}-G_{2,1},G_{1,3}-G_{2,2},\cdots\cdots,\)\(G_{1,\mu+1}-G_{2,\mu}]^{\mathsf{T}}\).