## 1 Introduction  
In deep drilling applications, unwanted vibrations can reduce reliability, lead to premature failures of components, reduce the drilling efficiency and increase nonproductive time. Around one-third of all failures in drilling applications are related to vibrations [1, 2]. The occurring vibrations are divided into axial, lateral and torsional vibrations according to their operating direction. Since the late 1960s, torsional vibrations have been the research focus [3, 4]. Torsional vibrations are divided into low- and high-frequency vibrations. Low-frequency vibrations like stick/slip correspond to the first torsional mode of the drilling system, with frequencies less than 1 Hz. These self-excited vibrations are distinguished according to their origins in bit and string-induced vibrations. Bit-induced vibrations originate from the bit-rock cutting process, while string-induced vibrations are caused by the contact between drilling tools and formation [5]. The bit rock interaction during drilling is a complex process with many uncertainties [6], which can be considered, in various ways, e.g., with hybrid uncertainties [7]. It has been shown that, for the torsional dynamic of the drill string, the energy input can be modeled using a nonlinear torque characteristic at the bit [8]. Due to improved and new measuring tools, high-frequency oscillations have been identified as the cause of numerous drill string failures and have been intensively investigated over the last years with studies [9, 10], simulations [8, 11] and experiments [12, 13]. Especially drilling in hard and dense formations leads to high-frequency torsional oscillations (HFTO) with frequencies between 50 Hz and 500 Hz [14, 15]. By assuming a resistance characteristic at the bit (torque characteristic) that decreases with the angular speed (revolutions per minute RPM) (Figure 1), a predictive criterion based on linearization at the operation point (\(\frac{\text{dTorque}}{\text{dRPM}}\))  
\[\text{S}_{\text{c},k}=-\frac{2\text{D}_{k}\omega_{0,k}}{\varphi_{k}^{2}}<\frac{ \text{dTorque}}{\text{dRPM}} \tag{1}\]  
is derived [8], wherein \(D_{k}\) represents the modal damping, \(\omega_{0,k}\) the natural angular frequency and \(\varphi_{k}\) the deflection of the mass normalized eigenvector of mode k.  
Additionally, a maximum amplitude at the bit (\(\dot{\varphi}_{bit}\)) that depends on the revolutions per minute (RPM) in the operating point  
\[\dot{\varphi}_{bit}\omega_{0,k}=\dot{\varphi}_{\text{bit}}=\dot{\varphi}_{ \text{RPM}}=\frac{2\pi}{60\frac{sec}{mn}}\text{RPM}\rightarrow\ \dot{\varphi}_{bit}=\frac{2\pi}{60\frac{sec}{mn}}\text{RPM}\frac{1}{\omega_{0,k}} \tag{2}\]  
is derived in [16] and agrees with the observations in [14] that no backward rotation occurs at the bit. This maximum amplitude is valid when one HFTO mode dominates the dynamic of the entire drill string [17; 18]. In [17], the interaction between stick/slip and HFTO is observed and [19] analyzed in detail. Using the information about stick/slip, HFTO and the torque characteristic (Figure 1) stability maps are derived [19] and new strategies to reduce vibrations by adjusting operational parameters like the drilling velocity (RPM) and the downhole force on the bit (in drilling industry: weight on bit (WOB)) are shown [5]. As neither the parameters nor the design of the drill string can be changed arbitrarily, new downhole tools must be developed to reduce critical vibrations. An isolator tool based on the principle of mechanical low-pass filtering is analyzed in testing, simulation and operation [13] and reduces the amplitude in specific areas of the BHA. As the isolator does not mitigate the critical vibration along the entire drill string, further mitigation strategies are necessary. Increasing the damping of a system is a common approach to reduce self-excited vibration [20; 21]. In [22], the effects of various friction dampers on the stability of critical HFTO modes are analyzed.