## 4 Description of the Test Bench and Experimental Approach
### Fluid Model Properties  
For a rectangular section, the damping factor is a function of the liquid's height and the tank's width. For the aforementioned expression, \(\omega_{\mathrm{In}}=(\mathrm{g}\ \xi_{\mathrm{In}}/\mathrm{Wtanh}(\xi_{ \mathrm{In}}\mathrm{h}/\mathrm{W}))^{1/2}\) of the liquid-free surface gets closer to being constant. A family of unit-step response curves  
Figure 5: Miniature drilling rig model for experimental analysis. (**a**) 1-Computer screen, 2-ccout-2013, 3-speed control junction box, 4-frame support, 5-vertical shaft, 6-motor, 7-disc, 8-tank, 9-granitic rock sample; (**b**) 10-probe sensors, 11-water; (**c**) 12-rub screw, 13-rub housing; (**d**) 14, 15–90\({}^{\circ}\) mounted sensors.  
From Figure 6a, it can be seen that an underdamped fluid system between 0.1 and 0.9 gets close more rapidly than a critically damped or overdamped system. When increasing the damping parameters, the amplitude of the hydrodynamic force is exponentially increased away from the equilibrium position. The hydrodynamic forces acting on the rectangular container experience harmonic motion by overshooting the equilibrium position rapidly and exhibiting the fastest unstable response. Overdamped hydrodynamic forces are sluggish in responding to any damping increment, as observed by the dashed curves. However, Figure 6b presents the dependence of fluid forces on the lateral excitation frequency. From the foregoing analysis of the hydrodynamic forces, the fluid force is either critically damped or overdamped monotonically at the same magnitude. The downhole vibration of a drill bit, such as axial, lateral, and torsional vibrations, can cause premature tool failure. To optimize the drilling performance and extend the life of the drilling tools, controlling or mitigating downhole vibration is essential [19; 20]. The natural frequencies of the five DoF kinematic chain are as follows: \(f_{01}=0\) Hz\({}_{f02}=4.0756\) Hz\({}_{f0X}=\)53 Hz\({}_{f}\), \(f_{0Y}=\) 53 Hz, and \(f_{0Z}=\)187 Hz. The fluid drill string parameters used in this study are listed as indicated in Table 1.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Drill String Parameter** & **Value and Unit** & **Bearing Stiffness** & **Value and Unit** \\ \hline Length of the shaft (L) & 570 mm & Bearing shaft stiffness (\(K_{o}\)) & \(7.35\times 10^{5}\) Nm\({}^{-1}\) \\ Shaft diameter (D) & 10 mm & Casing stiffness (\(K_{s}\)) & \(6.9\times 10^{7}\) N/m \\ Disc radius (\(Rd\)) & 0.15 m & Lateral damping ratio (\(\xi_{l}\)) & 0.0287 \\ Disk moment of inertia (\(J_{D}\)) & 0.186 kg.m\({}^{2}\) & Torsional damping (\(\xi_{T}\)) & 0.03 \\ Motor moment of inertia (\(J_{M}\)) & 10.36 kg.m\({}^{2}\) & **Disc** & **Value and Unit** \\ Shaft torsional damping (\(C_{T}\)) & 90 Nms/rad & & \\ Damping of the shaft (\(C_{0}\)) & 100.44 kg/s & Shaft–disc mass (\(M\)) & 16.845 kg \\ Friction coefficient (\(\mu\)) & 0.2 & Eccentricity mass (m\({}_{u}\)) & 0.25 kg \\ Rotor–casing clearance (\(\delta\)) & 20 \(\mu\)m & Mass eccentricity (\(e\)) & 0.0011 m \\ Radial clearance (\(\delta_{ax}\)) & 60 \(\mu\)m & Disc thickness (\(D_{disc}\)) & 25 m \\ \hline
**Inviscid Fluid Properties** & **Units** & **Tank Parameter** & **Value and Unit** \\ \hline Water (\(T^{\circ}\)) & 20 \({}^{\circ}\)C & Tank width (\(W\)) & 0.3 m \\ Fluid frequencies (\(\omega_{\mathrm{In}}\)) & 6.01 rad/s & Excitation amplitude (\(X_{0}\)/\(W\)) & 0.166 \\ Fluid density (\(\rho\)) & 1004 kg/m\({}^{3}\) & Axial force magnitude (\(P_{j}\)) & 50 kN \\ Shaft–fluid angular speed (\(\Omega\)) & 4.8522 rad/s & Static weight component (\(P_{o}\)) & 100 N \\ \hline \hline \end{tabular}
\end{table}
Table 1: Data parameters used in rotary drilling simulation.  
Figure 6: Dependence of the ratio of the hydrodynamic forces on the damping and frequency parameters. (**a**) Fluid forces for various values of \(\xi_{l}\) (**b**) Fluid forces around the oscillating drill.