## 2 Materials and Methods  
The method used for designing interference fits considers the contact pressure, \(p\), uniformly distributed at the interface shaft/hub [6; 7]. This pressure can be estimated by means of the following equation when the shaft and hub are composed of different materials:  
\[p=\frac{\delta}{\frac{r}{E_{\text{o}}}\left(\frac{R^{2}+r^{2}}{R^{2}-r^{2}}+ \
u_{\text{o}}\right)+\frac{r}{E_{i}}(1-\
u_{i})} \tag{1}\]  
where \(\delta\) is the radial interference, \(r\) is the radius at the interface shaft/hub, \(R\) is the outer radius of the hub, \(E_{\text{o}}\) and \(\
u_{\text{o}}\) are the Young modulus and Poisson coefficient of the hub, respectively, and \(E_{\text{i}}\) and \(\
u_{\text{i}}\) are the Young modulus and Poisson coefficient of the shaft, respectively. When the hub and shaft are composed of the same material, the previous equation can be simplified as follows:  
\[p=\frac{\delta}{2}\frac{E}{r}\left(1-\frac{r^{2}}{R^{2}}\right) \tag{2}\]  
FEM Simulations were carried out with a commercial code (MSC.MARC) considering as a reference case an interference fit of a hard press type named 200H7/s6 according to the ISO standard for fits and tolerances [26]. This fit is recommended by such a standard for cases where a fixed positioning of the shaft and hub during rotation movement is required for transmitting both movement and a torque. On one hand, the dimensions of the interference fit used as a reference case were taken according to the ratios given  
Figure 1: Scheme of a shrink fit using a contact ring. by previous studies [3; 19]. Thus, the shaft dimensions (radius, \(r\) and length \(l\)) and hub dimensions (radius \(R\) and length \(L\)), were selected according to the ratios \(l/L\)= 4, and \(R/r\) = 1.2 [19]. Accordingly, the interference fit taken as a reference case has a shaft length of \(l\) = 400 mm, a radius of \(r\) = 200 mm, a hub radius of \(R\) = 240 mm, and a hub length of \(L\) = 100 mm. Taking this into account, the resulting interference closure is \(\delta\) = 75.5 \(\upmu\)m. On the other hand, different cases were considered varying the contact ring thickness (\(t_{r}\)) considering for all cases analyzed the same thickness of the hub (including the contact ring) of \(t\) = 40 mm. Accordingly, the six cases of study are listed as follows in terms of the dimensionless thickness ratio defined as \(\gamma\)= \(t_{r}/t\): (i) a 2.5% of the total hub thickness (\(t\)), \(\gamma\) = 0.025; (ii) a 5% of \(t\), \(\gamma\) = 0.050; (iii) a 12.5% of \(t\), \(\gamma\) = 0.125; (iv) a 17.5% of \(t\), \(\gamma\) = 0.1750; (v) a 25% of \(t\), \(\gamma\) = 0.250; and finally (vi) a half of the total hub thickness, \(\gamma\) = 0.500. Due to the revolute symmetry of both the shaft and hub (including the contact ring), the modeling can be simplified to an axisymmetric case in the way shown in Figure 2.  
This way, the transversal symmetry axis #1 in Figure 2 was used for simplifying the geometry to half by imposing a null displacement in the \(x\)-axis direction at points placed on such an axis. On the other hand, symmetry axis #2 allows one to reduce to half by imposing null displacements in the \(y\)-axis direction to the points placed on such an axis. Considering  
Figure 2: Scheme of the simplified geometry and boundary conditions applied in FEM simulations. the aim of the study, the main attention was paid to the interface contact ring/shaft. Thus, the following contact conditions were imposed. On one hand, the contact hub/contact ring was assumed to be perfectly united, and on the other hand, the interference closure at the interface shaft/contact ring corresponding to the selected interference fit was included in the FEM simulations by means of an option available in the FEM code MSC.MARC when the contact type (named \"touching\") imposed between the shaft and contact ring is selected. This option allows one to impose a given radial interference causing contact pressure at the interface of the interference fit simulated.