## 3 Results
The cross-coupled damping coefficient \(D_{tx,ty}\) does not exceed 1 kN m s/rad in magnitude, being mostly negative (see Figure 9). \(D_{tx,ty}\) rapidly decreases with decreasing \(t_{x}\) or increasing \(t_{y}\) in the negative \(t_{x}\) range, while in the positive \(t_{x}\) range, it is much less sensible both to \(t_{x}\) and \(t_{y}\) (see Figure 9a). \(D_{tx,ty}\) decreases with increasing \(t_{y}\) and is almost insensible to \(\dot{t}_{y}\) (see Figure 9b). The dependence of \(D_{tx,ty}\) on eccentricity parameters is highly nonlinear. \(D_{tx,ty}\) can take relatively high positive values for zero \(\varepsilon_{x}\). and positive \(\varepsilon_{y}\), but in most of the range, the sensitivity to \(\varepsilon_{x}\) and \(\varepsilon_{y}\) is low (see Figure 9c). Figure 8: Parameter sensitivity of the direct damping coefficient \(D_{tx,tx}\): (**a**) Effects of the tilting angle and tilting velocity in the x-direction; (**b**) Effects of the x and y tilting angles; (**c**) Effects of the relative eccentricities. All unreferenced parameters have zero values. The cross-coupled damping coefficient \(D_{ty,tx}\) falls in the range \(-0.6\)-\(0.13\) kN m s/rad (see Figure 10). It is negative for positive \(t_{y}\) and negative \(t_{x}\), otherwise it is positive. Similar to \(D_{tx,ty}\), \(D_{ty,tx}\) rapidly decreases with decreasing \(t_{x}\) or increasing \(t_{y}\) in the negative \(t_{x}\) range, while in the positive \(t_{x}\) range, it is much less sensible both to \(t_{x}\) and \(t_{y}\). However, for \(t_{x}>0.002\ ^{\circ}\), \(D_{ty,tx}\) increases with increasing \(t_{y}\) (see Figure 10b), while \(D_{tx,ty}\) decreases with increasing \(t_{y}\) for any \(t_{x}\) (see Figure 9a). For zero \(t_{y}\), \(D_{ty,tx}\) decreases with increasing \(t_{x}\) and decreasing \(t_{x}\). Similar to all previously considered damping coefficients, the dependence of \(D_{ty,tx}\) on the eccentricity parameters is highly nonlinear. Figure 9: Parameter sensitivity of the cross-coupled damping coefficient \(D_{tx,ty}\): (**a**) Effects of x and y tilting angles; (**b**) Effects of the tilting angle and tilting velocity in the y-direction; (**c**) Effects of relative eccentricities. All unreferenced parameters have zero values. In contrast with all other damping coefficients, the direct damping coefficient \(D_{ty,ty}\) does not significantly depend on \(\varepsilon_{x}\) and \(\varepsilon_{y}\) (refer to Figure 11). It falls in the range 0.6-2.0 kN\(\cdot\)m s/rad. It is higher for lower \(t_{x}\), and higher \(t_{y}\), similar to \(D_{tx,tx}\). \(D_{ty,ty}\) is almost insensible to the y-tilt velocity \(\dot{t}_{y}\). It slightly increases with increasing \(\varepsilon_{x}\) and increasing \(\varepsilon_{y}\). Figure 10: Parameter sensitivity of the cross-coupled damping coefficient \(D_{ty,tx}\): (**a**) Effects of the tilting angle and tilting velocity in the x-direction; (**b**) Effects of x and y tilting angles; (**c**) Effects of relative eccentricities. All unreferenced parameters have zero values. In summary, the direct stiffness and damping coefficients were positive. The cross-coupled stiffness coefficients are approximately 10 times smaller than the direct stiffness coefficients, and the cross-coupled damping coefficients are approximately 2-3 times smaller than the direct damping coefficients. Hence, a stable tilting motion was expected. However, information on the rotor inertia and the bearing at the other end is required to make precise conclusions on the dynamic behavior. Moreover, the results above exhibit a strong coupling between the eccentricity and tilting motion, including the nonlinearity of the moments relative to the position parameters. This can make the rotor motion severely complicated. Figure 11: Parameter sensitivity of the direct damping coefficient \(D_{ty,yy}\): (**a**) Effects of x and y tilting angles; (**b**) Effects of the tilting angle and tilting velocity in the y-direction; (**c**) Effects of relative eccentricities. All unreferenced parameters have zero values.