## 6 Transfer to Self-Excited Drill String Vibrations
### Comparison with Time Domain Simulations  
The damping diagram in Figure 9a is derived by determining the equivalent damping ratio (Equation (52)) from the analytical solution (Equation (23)) for one natural frequency of the underlying self-excited structure (excitation frequency of the analytical solution). The natural frequency of the underlying structure is higher than the linear natural frequency of the damper to show the results regarding the nonlinear stiffness damper. In addition to the damping ratio provided by the damper, Figure 9a shows the absolute values of three different self-excited damping ratios. For two damping ratios \(D_{i}=-1\%\) and \(D_{i}=-1.6\%\) intersections occur between the self-excited absolute damping ratios and the damping provided by the damper. The highest negative damping ratio (\(D_{i}=-2.1\%\)) exceeds the maximum additional damping provided by the damper. Figure 9b shows the relative displacement corresponding to the damper with respect to the amplitude of the underlying structure and the characteristic relative displacement (Equation (54)). For the smallest instability--thus, a linearized negative damping ratio of \(D_{i}=-1\%\)--the self-excited oscillation is limited and, thus, stabilized by the damping effect of the nonlinear damper. Due to the nonlinear stiffness and the frequency difference between the linear natural frequency of the damper and the vibration frequency of the structure, the damping effect is sufficient to stabilize the system above a certain amplitude. Therefore, at small amplitudes, the amplitude increases exponentially, as usual in self-excited systems (Figure 10a). At a certain amplitude, the damping ratio provided by the damper is sufficient to stabilize the self-excited system. The vibration amplitude of the structure is similar to the amplitude of the intersection in Figure 9a. As shown in Figure 8b the characteristic frequency and, thus, the damping is increased near the characteristic relative displacement. Due to the small, required damping ratio, the relative displacement does not exceed the characteristic relative displacement (Figure 10b). Figure 8: Damper with a cubic nonlinear stiffness (\(\alpha=7\cdot 10^{11}\)\(Nm,d=50\)\(\frac{Nm}{s}\), \(\omega_{0}=1000\)\(\frac{1}{s}\), \(\Omega=1250\)\(\frac{1}{s}\), \(J=0.2\)\(kgm^{2}\)). (**a**) Relative angular displacement and (**b**) equivalent damping ratio over the excitation frequency (natural frequency of the self-excited system) and angular displacement of the underlying structure (excitation frequency of the analytical system). Figure 9: Damper with a cubic nonlinear stiffness (\(\alpha=7\cdot 10^{11}\)\(Nm,d=50\)\(\frac{Nm}{s}\), \(\omega_{0}=1000\)\(\frac{1}{s}\), \(\Omega=1250\)\(\frac{1}{s}\), \(J=0.2\)\(kgm^{2}\)). (**a**) Equivalent damping ratio regarding the angular displacement of the underlying structure. (**b**) Relative angular displacement regarding the angular displacement of the underlying structure. For higher self-excitations and, thus, higher negative damping ratios (\(D_{i}=-1.6\%\)), the intersection occurs at higher damping ratios. Figure 11a shows similar to Figure 10a at first an exponential increase of the amplitude at small relative displacements between the damper and the structure. The characteristic frequency of the nonlinear damper for this amplitude is below the vibration frequency of the structure. Due to the self-excitation, the amplitude, as well as the relative displacement between the damper and structure, increases. This increase of the relative displacement changes the characteristic frequency of the damper and, thus, increases its damping effect. This increased damping results in a reduction of the vibration amplitude of the structure. The smaller amplitude of the structure results in a smaller relative displacement and, thus, in a reduced damping effect. This repetitive increase and decrease of the amplitude and damping ratio results in a nonperiodic attractor (Figure 11a).