## 6 Transfer to Self-Excited Drill String Vibrations
### Comparison with Time Domain Simulations
As Figure 11b shows, the relative displacement amplitude occurs around the characteristic relative amplitude. Similar to the self-excitation with \(D_{i}=-1\%\), the amplitude of the structure is similar to the amplitude of the intersection in Figure 9a. For even higher self-excitations, e.g., \(D_{i}=-2.1\%\), the additional damping is not sufficient to stabilize the unstable mode. Thus, an exponential increase of the amplitude occurs (Figure 12a,b). In  
Figure 11: Time domain results for an instability of \(D_{i}=-1.6\%\). (**a**) Angular displacement of the drill string structure and the damper. (**b**) Relative angular displacement between the damper and the drill string structure. Figure 10: Time domain results for an instability of \(D_{i}=-1\%\). (**a**) Angular displacement of the drill string structure and the damper. (**b**) Relative angular displacement between the damper and the drill string structure. Figure 12, a change in the dynamic motion of the structure regarding the characteristic frequency is visible at \(t\approx 0.2s\). Figure 9a shows that the analytical results can be easily applied to self-excited structures. For self-excitations near the maximum of the damping ratio provided by the damper (e.g., \(-1.9\%\) in Figure 9a), the system is unstable, although according to the analytical solution, the system should be stabilized. Reasons for this small inaccuracy are the assumptions mentioned in Sections 2 and 3, like the constant harmonic oscillation of the structure, the negligible reactive effect of the damper on the structure and the order of MSLP. One additional and very important reason is the combination of multiple possible occurring solitons in combination with the energy input and, thus, change of the amplitude in self-excited structures. This effect is shown in Figure 11. The damping effect is increased when the relative angular displacement is close to the characteristic relative angular displacement. The necessary relative displacement is achieved when the amplitude of the structural vibration increases. This is seen well at higher natural frequencies of the self-excited structure, where the influence of the nonlinearity is high. Figure 13a shows the damping diagram when the natural frequency of the structure is increased to \(1500\frac{1}{s}\). Figure 13b shows that the instabilities of \(D_{i}=-0.8\%\) still resulted in a stable attractor due to the damping provided by the damper (similar to Figure 11). Figure 13a shows additionally to the damping diagram a principal depiction of the \"closed loop\" process, resulting in the attractor (Figure 13b). For a self-excited structure without significant shocks, the amplitude increases exponentially from a small initial amplitude. Due to the cubic stiffness nonlinearity of the system, the relative displacement and the corresponding damping ratio is low at small amplitudes and, thus, leads to an increase of the amplitude (Figure 13a arrow 1). At some point, an additional stable solution occurs, but the provided damping is low due to the initial small relative displacement (Figure 13a arrow 1). At higher amplitudes, the solution with small relative displacements disappears, and a rapid increase of the relative displacement between damper and structure leads to an increase of the provided damping (Figure 13a arrow 2). In Figure 13, the increased damping is sufficient to stabilize the instable mode and results in a decrease of the amplitude of the structure (Figure 13a arrow 3). This decrease of the structural amplitude leads to an increase of the provided damping and, thus, to an even faster decrease of the amplitude. Finally, the amplitude decreases below the amplitude of the maximum possible damping ratio, which causes the solution with high relative displacements and damping ratios to disappear (Figure 13a arrow 4). Due to the small relative displacement and damping ratio a repeated increase of the amplitude  
Figure 12: Time domain results for an instability of \(D_{i}=-2.1\%\).