## 3 Algorithm Considering Centrifugal Force and Temperature Gradient
### Temperature Gradient  
Temperature distribution function is shown as [19]:  
\[t=t_{a}+\frac{t_{b}-t_{a}}{\ln(r_{2}/r_{1})}\ln(\rho/r_{1}) \tag{12}\]When the cylinder is expanded by heat, the displacement of any point is shown as:  
\[u_{i}=\int_{0}^{\rho}\alpha td\rho \tag{13}\]  
According to Equations (12) and (13), the displacement due to the thermal expansion of cylinder can be obtained:  
\[u_{i}=\alpha\rho[t_{a}+\frac{t_{b}-t_{a}}{\ln(b/a)}(\ln\frac{\rho}{a}-1)] \tag{14}\]  
where \(\alpha\) is the coefficient of thermal expansion, \(t_{a}\) is the temperature of inner surface of cylinder, and \(t_{b}\) is the temperature of outer surface of cylinder. Assuming that \(Q_{n,i}\) is radial displacement due to temperature, \(t_{n,i}\) is cylinder surface temperature. Substituting \(C_{i}\) parameter into Equation (14), the radial displacement of outer surface of cylinder \(C_{i}\) under the temperature can be obtained:  
\[Q_{2,i}=\frac{1}{2}\alpha d_{2,i}\bigg{(}t_{2,i}+\frac{t_{1,i}-t_{2,i}}{\ln d _{2,i}-\ln d_{1,i}}\bigg{)} \tag{15}\]  
Similarly, the radial displacement of the inner surface of cylinder \(C_{i+1}\) under the temperature is shown as:  
\[Q_{1,i+1}=\frac{1}{2}\alpha d_{1,i+1}\bigg{(}t_{1,i+1}+\frac{t_{1,i+1}-t_{2,i +1}}{\ln d_{2,i+1}-\ln d_{1,i+1}}\bigg{)} \tag{16}\]  
The displacement of contact surface of multi-layer interference cylinder under temperature gradient is shown as:  
\[ \begin{array}{ll}\Delta Q_{i}&=Q_{1,i+1}-Q_{2,i}\\ &=\frac{1}{2}\alpha d_{1,i+1}\bigg{(}\frac{\Delta t_{i+1}}{\ln d_{2,i+1}-\ln d _{1,i+1}}-\frac{\Delta t_{i}}{\ln d_{2,i}-\ln d_{1,i}}\bigg{)}\end{array}  \tag{17}\]  
According to Equations (2) and (17), the matrix model of interference magnitude of multi-layer interference cylinder under the temperature gradient is shown as:  
\[\delta=Kp+\Delta Q \tag{18}\]  
And, \(\Delta Q=[Q_{1,2}-Q_{2,1},Q_{1,3}-Q_{2,2},\cdots\cdots,Q_{1,n+1}-Q_{2,n}]^{ \mathrm{T}}\). **4. Finite Element Model**  
Here, five cylinders named by \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(C_{5}\) are constructed with steel material, and the multi-layer interference cylinder model is consists of four interference cylinders named by \(M_{1}\), \(M_{2}\), \(M_{3}\), \(M_{4}\). The according parameters of models are listed in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cylinder No** & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) \\ \hline Internal diameter (mm) & 100 & 200 & 300 & 400 & 500 \\ External diameter (mm) & 200 & 300 & 400 & 500 & 600 \\ Height (mm) & & & 150 & & \\ Elastic modulus & & & 206 GPa & & \\ Poisson ratio & & & 0.3 & & \\ Density & & & 7800 kg/m\({}^{3}\) & & \\ Coefficient of linear expansion & & & \(1.1\times 10^{-5}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters. According to National Standard GB/T1800.4-1999, the maximum interference magnitude between \(C_{1}\) and \(C_{2}\) is 0.195 mm, \(C_{2}\) and \(C_{3}\) is 0.272 mm, \(C_{3}\) and \(C_{4}\) is 0.330 mm, \(C_{4}\) and \(C_{5}\) is 0.400 mm. Model \(M_{1}\), \(M_{2}\), \(M_{3}\), \(M_{4}\) is as shown in Figure 4.  
By Equations (11) and (18), the contact pressure of each surface of multi-layer interference cylinder based on centrifugal force and temperature gradient can be respectively obtained. To verify its correctness and analyze effects of centrifugal force and temperature gradient on the contact pressure, three-dimensional models of \(M_{1}\), \(M_{2}\), \(M_{3}\), and \(M_{4}\) are established by finite element software ABAQUS 6.14. A \"penalty-type\" friction formulation was applied to each interface, and the friction coefficient is set as 0.15. Figure 5a is boundary condition for \(M_{4}\) as an example, which is used to fix displacement (U1 = U2 = U3 = 0) of cylinder \(C_{1}\) and apply rotational body force, then the different speed gradients are set up to obtain numerical solution of contact pressure of multi-layer interference cylinder under centrifugal force. Figure 5b is a model with three-dimensional meshes [20].