## 2 Analytical Solution of Multiple Contact Pairs for the MLTWC
### General Analytical Solution of Multiple Contact Pairs for the MLTWC  
Without loss of generality, the interference fit of a MLTWC system with \(n\) thick-walled cylinders is studied, as shown in Figure 8. A holistic coupling method is developed to analyze the interference of MLTWC. The calculation flow diagram for the interference algorithm without consideration of assembly sequence is illustrated in Figure 9.  
The interference fit between the \(i^{th}\) layer and \((i+1)^{th}\) layer of the MLTWC is shown in Figure 8, which is composed of the outer surface of the \(i^{th}\) thick-wall cylinder and the inner surface of the \((i+1)^{th}\) cylinder. The radial displacement \(u\) of these two surface needs to satisfy the displacement coordination described in Equation (29), so that there is no gap or penetration between these two surfaces. \[u_{i+1}^{in}-u_{i}^{out}=\delta_{i,i+1}\left(i=1,2,\ldots,n-1\right) \tag{29}\]  
where the superscript represents the inner and outer surface, respectively. Figure 8: Sketch of interference fit of MLTWC. Under the hypothesis of small elastic deformation, the interference fit of n MLTWC satisfies superposition principle, which can be equivalent to \(n\) axisymmetric thick-wall cylinders with uniformly distributed interior and exterior pressure. According to the elastic theory solution, the radial displacement of the outer surface of the \(i^{th}\) cylinder is shown in Equation (30). \[u_{i}^{out}=\frac{R_{i}\left[P_{i-1}\left(R_{i-1}-\delta_{i-1,i}\right)^{2}-P_{ i}R_{i}^{2}\right]\left(1-v_{i}\right)}{E_{i}\left(R_{i}^{2}-\left(R_{i-1}- \delta_{i-1,i}\right)^{2}\right)}+\frac{\left(P_{i-1}-P_{i}\right)\left(R_{i-1 }-\delta_{i-1,i}\right)^{2}R_{i}\left(1+v_{i}\right)}{E_{i}\left(R_{i}^{2}- \left(R_{i-1}-\delta_{i-1,i}\right)^{2}\right)} \tag{30}\]  
Since \(\delta_{i-1,i}\) is small compared to \(R_{i-1}\), Equation (30) can be simplified as in Equation (31). \[u_{i}^{out}=\frac{R_{i}\left[P_{i-1}R_{i-1}^{2}-P_{i}R_{i}^{2}\right]\left(1-v _{i}\right)}{E_{i}\left(R_{i}^{2}-R_{i-1}^{2}\right)}+\frac{\left(P_{i-1}-P_{ i}\right)R_{i-1}^{2}R_{i}\left(1+v_{i}\right)}{E_{i}\left(R_{i}^{2}-R_{i-1}^{2}\right)} \tag{31}\]  
Similarly,  
\[u_{i+1}^{in}=\frac{R_{i}\left(P_{i}R_{i}^{2}-P_{i+1}R_{i+1}^{2}\right)\left(1- v_{i+1}\right)}{E_{i+1}\left(R_{i+1}^{2}-R_{i}^{2}\right)}+\frac{\left(P_{i}-P_{ i+1}\right)R_{i}R_{i+1}^{2}\left(1+v_{i+1}\right)}{E_{i+1}\left(R_{i+1}^{2}-R_{i}^{2} \right)} \tag{32}\]  
Put Equations (31) and (32) into Equation (29), then  
\[\delta_{i,i+1}=R_{i}\left(\frac{2P_{i-1}R_{i-1}^{2}-P_{i}\left(-R_{i-1}^{2} \left(-1+v_{i}\right)+R_{i-1}^{2}\left(1+v_{i}\right)\right)}{E_{i}\left(R_{i- 1}^{2}-R_{i}^{2}\right)}+\frac{2P_{i+1}R_{i+1}^{2}-P_{i}\left(R_{i+1}^{2}+R_{ i+1}^{2}\right)+P_{i}\left(R_{i}^{2}-R_{i+1}^{2}\right)\left(v_{i+1}\right)}{E_{i+1} \left(R_{i}^{2}-R_{i+1}^{2}\right)}\right) \tag{33}\]  
Expand it and group terms, Equation (34) can be derived. Figure 9: Calculation flow diagram for interference algorithm for multi-layer thick-walled cylinders (MTC) without consideration of assembly sequence. \[\delta_{i,i+1}=\frac{2R_{i-1}^{2}R_{i}}{E_{i}\left(R_{i-1}^{2}-R_{i}^{2}\right)}P_{ i-1}+[\frac{R_{i}(R_{i}^{2}+R_{i-1}^{2})}{E_{i}\left(R_{i}^{2}-R_{i-1}^{2}\right)}- \frac{R_{i}v_{i}}{E_{i}}+\frac{R_{i}(R_{i+1}^{2}+R_{i}^{2})}{E_{i+1}\left(R_{i+1 }^{2}-R_{i}^{2}\right)}+\frac{R_{i}(v_{i+1})}{E_{i+1}\left(R_{i}^{2}-R_{i+1}^{2 }\right)}P_{i}+ \tag{34}\]  
It can be seen from Equation (34) that \(P_{i-1}\), \(P_{i}\), and \(P_{i+1}\) all have their own contributions to \(\delta_{i,i+1}\), which is determined by the parameters of material and geometrical characteristics. Represent the coefficients of \(P_{i-1}\), \(P_{i}\), and \(P_{i+1}\) in Equation (34) by \(S_{i,i-1}\), \(S_{i,i}\), and \(S_{i,i+1}\), respectively, and Equation (34) can be rewritten as:  
\[\left\{