## 2 Mathematical Modeling of the Drill String System
### Mathematical Modeling of the 3D Rubbing Contact  
During the experiment, the drill string, influenced by a centrifugal force during bending, always touched the borehole wall during drilling, and the borehole wall exerted pressure and frictional force on the drill string in response. As a result, two points of contact (axial bit load, shaft disc to casing wall) could, in the long run, cause premature wear and cracks. The collision effects between the shaft disc and the fixed casing are simplified to be the axial and lateral impact forces exerted on the outer surface of the casing, as shown schematically in Figure 2a. Figure 2b illustrates how the model proposed here, in contrast to others in the reviewed literature, accounts for the shaft's longitudinal sliding in the vertical direction \(Z\) to estimate the nonlinear forces during contact. It is suggested that the expression of the rubbing forces between rolling parts and the casing can be used to create a realistic rubbing model that is more complex than the typical radial bilinear model [16]. Although various friction schemes have been proposed, such as the Stribeck friction model, one of the most extensively used models to explain the friction phenomenon is the Coulomb friction law. In the cartesian coordinate system of X, Y, and Z (see Figure 2b), let S represent any known site of disc and casing contact. The axis Z is chosen to be orientated along the drill string's normal, while axes X and Y are oriented along the main axes of the cross-section. Let, \(\overrightarrow{e}\), \(\overrightarrow{e}_{Y}\), and \(\overrightarrow{e}_{Z}\) be the unit vectors in the three-dimensional space X-, Y-, and Z-axes, respectively. The disc bends in the \(\overrightarrow{e}_{X}\) and \(\overrightarrow{e}_{Y}\) directions and also oscillates along the direction of \(\overrightarrow{e}_{Z}\) (Figure 2a). The current position of the contact point \(S\) with respect to \(O\) as shown in Figure 3b is written by the vector. The contact then brings an added stiffness that is higher than the shaft stiffness and induces a restoring force. The expression of the radial impact force \(F_{N}\) and the tangential rub force \(F_{T}\) can be written by the Coulomb-Amontons law for friction as:  
\[ \begin{cases}F_{N}=(\Delta-\delta)K_{s},&for\ \ \Delta\geq\delta\\ F_{T}=\mu F_{N},&for\ \ \Delta<\delta\end{cases}  \tag{6}\]  
where \(K_{s}\) is the stiffness of the casing in the radial direction, \(\Delta=\sqrt{\left(S_{X}^{r}-S_{X}^{s}\right)^{2}+\left(S_{Y}^{r}-S_{Y}^{s} \right)^{2}}\), which represents the radial relative displacement between the rotor (indexed by \(r\)) and the casing (indexed by \(s\)), \(\mu\) is Coulomb's friction coefficient, and \(\delta\) is the radial clearance between the rotor and the casing. \(S_{X}\) and \(S_{Y}\) are, respectively, used for horizontal displacements and \(S_{Z}\) for vertical displacement. The mathematical phenomenon behind the nonlinearity due to the rubbing effect happens when the radial displacement becomes equal to the clearance between the drill string and casing (\(\Delta=\delta\)). The forces exerted in the plane between the drill string and the casing are therefore expressed as:  
\[ \begin{cases}F_{X}=-\cos\phi.F_{N}+\sin\phi.F_{T}\\ F_{Y}=-\sin\phi.F_{N}-\cos\phi.F_{T}\end{cases} \qquad\text{with}\qquad \begin{cases} \cos\phi=\left(S_{X}^{r}-S_{X}^{s}\right)/\Delta\\ \sin\phi=\left(S_{Y}^{r}-S_{Y}^{s}\right)/\Delta^{\prime}\end{cases}  \tag{7}\]  
This model is more generally used to model the rotor/casing contact in a Cartesian plane [16].