## 2 Materials and Methods  
### Analytical Approach  
Let us consider the 2D model of an RB pivot in Figure 2.  
Here, \(O_{h}\) is the housing center, \(O_{p}\) is the center of the pad bore, \(O_{b}\) is the center of the pad back surface or pivot surface. \(O_{p}\) coincides with \(O_{h}\) when preload is zero and tilt angle is zero. In Figure 2a, the theoretical contact point is \(P_{1}\) which coincides with \(B_{1}\). In Figure 2b, the pad tilts by an angle:  
\[\alpha_{t}=\alpha_{b}-\alpha_{h} \tag{1}\]  
where \(\alpha_{b}=\widehat{B_{1}O_{b}B_{2}}\) and \(\alpha_{h}=\widehat{P_{1}O_{h}P_{2}}\). In Figure 2b, the theoretical contact point is \(P_{2}\) which coincides with \(B_{2}\). It is assumed that the pad tilts without sliding. Let \(\xi\) and \(\eta\) be a reference system fixed with respect to the pad. Neglecting viscous forces, the resultant of a generic pressure distribution is a force \(F\) passing through the point \(O_{p}\). In addition, to satisfy moment equilibrium of the pad, the resultant \(F\) must pass through the theoretical pivot point \(P_{2}\). Therefore, the angle \(\gamma\) between the resultant \(F\) and the axis \(\xi\) is different from zero. The computed pressure distribution should produce a null resultant moment with respect to point \(P_{2}\), with this point depending on the tilting angle. However, the used bearing calculation software does not consider the motion of the theoretical contact point; in particular, the resultant moment is computed with respect to point \(B_{1}\). In Figure 2b, the resultant moment of hydrodynamic forces with respect to the point \(B_{1}\) is:  
\[M=\big{(}R_{p}+t\big{)}F_{\eta} \tag{2}\]  
where \(R_{p}=O_{p}A\) is the pad bore radius, \(t=AB_{1}\) is the pad thickness, \(F_{\eta}\) is the component of the force \(F\) along \(\eta\). The displacement of the theoretical contact point can thus be included into the program by introducing an equivalent rotational stiffness \(K_{rot}\), such that:  
\[\big{(}R_{p}+t\big{)}F_{\eta}=K_{rot}\alpha_{t} \tag{3}\]  
Figure 2: Schematics of an RB pivot. (**a**) no tilt; (**b**) finite tilt. Blue lines are construction, axis, dimension and extension lines, the red arrow indicates the resultant force on the pad. The angle \(\gamma\) satisfies the following geometrical relation:  
\[\tan\gamma=\frac{R_{b}\sin\alpha_{b}}{R_{b}\cos\alpha_{b}+R_{p}+t-R_{b}} \tag{4}\]  
where \(R_{b}=O_{b}B_{2}=O_{b}B_{1}\) is the pad back radius. Moreover:  
\[\tan\gamma=F_{\eta}/F_{\xi} \tag{5}\]  
where \(F_{\xi}\) is the force component along \(\xi\). In addition, the rolling without sliding motion imposes:  
\[R_{b}\alpha_{b}=R_{h}\alpha_{h} \tag{6}\]  
where \(R_{h}=O_{h}P_{2}=O_{h}P_{1}\) is the housing radius. Therefore, after some manipulation of the above Equations, \(K_{rot}\) can be written as:  
\[K_{rot}=F_{\xi}\frac{\big{(}R_{p}+t\big{)}R_{b}R_{h}\sin\alpha_{b}}{\big{(}R_{ h}-R_{b}\big{)}\big{(}R_{b}\cos\alpha_{b}+R_{p}+t-R_{b}\big{)}\alpha_{b}} \tag{7}\]  
which, for small tilting angles, reads:  
\[K_{rot}\cong F_{\xi}\frac{R_{h}R_{b}}{R_{h}-R_{b}} \tag{8}\]  
The pad back is here assumed circular for the sake of simplicity; however, if it were not circular, the procedure is generalizable by employing in Equation (8) a pad back radius \(R_{b}(\alpha_{t})\) as a function of the tilting angle. Equation (8) is worth some additional discussion. Firstly, the hypothesis of small tilting angle is typically reasonable in real tilting pad journal bearings, as the tilting angles are of the order of \(10^{-3}\) radians. Then, the equivalent rotational stiffness is proportional to the equivalent radius of the contact and to the pad normal load, which can be computed as:  
\[\alpha_{t}=\alpha_{b}-\alpha_{h} \tag{1}\]0  
where \(p(\theta,z)\) is the pressure field, \(z\) is the axial coordinate, and \(\theta\) is the circumferential coordinate, with \(\theta=0\) corresponding to the \(\xi\) axis.