## 3 [PERSON] Tuning of a Single TSBD Attached to an SDOF-Main System  
To achieve the optimum damping effect of the single attached TSBD attached to an SDOF-main system, the two design parameters \(\delta=\omega_{TSBD}/\omega_{H}=f_{TSBD}/f_{H}\) and \(\zeta_{TSBD}\) must be selected appropriately. \(\delta\) and \(\zeta_{TSBD}\) define the ratio of the natural frequency of the TSBD to the natural frequency of the SDOF-main system and the linearized viscous damping of the TSBD. The analytical formulas for determining the optimum design parameters under time-harmonic force excitation \(F(t)=F_{0}\cdot cos(\Omega t)\) of an SDOF-main system, where \(F_{0}\) and \(\Omega\) are the excitation force amplitude and the circular excitation frequency, and under the conservative assumption of an undamped main system, were first derived by [PERSON] [4],  
\[\delta_{opt}=\ \frac{\omega_{TSBD}}{\omega_{H}}\ \ =\ \ \frac{f_{TSBD}}{f_{H}}\ \ =\ \ \frac{1}{1+\mu}\,\ \zeta_{TSBD,opt}=\ \ \sqrt{\ \frac{3\mu}{8\ \left(1+\mu\right)^{3}}} \tag{14}\]  
Equation (14) shows that the optimum design parameters of the harmonic force excited the SDOF-main system with a single TSBD attached depend solely on the mass ratio \(\mu=m_{R}/m_{H}\). In practical applications of the TSBD, the mass ratio is chosen in the range of 0.5 to 5%, and, therefore, the optimum frequency ratio results in 0.995-0.952. For instance, in the case of a natural frequency of the SDOF-main system of \(f_{H}=1.0\) Hz and \(\mu=5\%\), the optimum natural frequency of the vibration damper results in \(f_{TSBD}=0.95\) Hz. The corresponding optimal value of the Lehr damping ratio, in this case, would be \(\zeta_{TSBD}=0.127=12.7\%\). A deviation from the optimal design parameters, which, for example, may result from a change in the natural frequency of the main system over its lifetime, leads to a decrease in the effectiveness of the vibration damper. [PERSON] [1] shows that deviations from the optimal natural frequency of the vibration absorber leads to significantly higher losses of effectiveness compared to deviations from the optimal Lehr damping ratio. For instance, a detuning of the vibration damper's eigenfrequency by 15% leads to a tripling of the maximum vibration response of the main system, for \(\mu=4\%\). In contrast, an increase of the Lehr damping ratio by the factor of 1.5 leads to an insignificant loss of the optimum vibration absorber effect, for \(\mu\) = 4%. The losses in the effectiveness of not optimally tuned vibration absorbers become smaller at higher mass ratios, i.e., the damper is less prone to detuning that occurs naturally during its service life. To counteract the detuning effect of vibration dampers, the application of semi-active dampers proves to be advantageous. Hence, an innovative approach for the extension of the passive TSBD to a semi-active damping device is presented in Section 6.  
It is noted that the optimal design parameters given in Equation (14) are applicable in the case of a harmonic force-excited SDOF-main system and under the assumption of an undamped main system \(\zeta_{H}\) = 0. For other possible excitations and response parameters of an SDOF-main system, the optimum design parameters are given in [1,33]. An extension of [PERSON]'s tuning formulas to Multi-Degree-Of-Freedom (MDOF) main systems are presented in [34] and a state space optimization approach for MDOF-main systems and multiple attached vibration absorbers is given in [14,15]. In addition, [PERSON] and [PERSON] [35] give the following analytical equations for the correction of the optimal design parameters considering the structural damping of the main system \(\zeta_{H}\) > 0 when optimizing the vibration damper,  
\[\begin{array}{l}\widetilde{\delta}_{opt}\ =\ \delta_{opt}\ -\ \left(0.241+1.7\ \mu-2.6\ \mu^{2} \right)\ \zeta_{H}\ -\ \left(1-1.9\ \mu+\mu^{2}\right)\ \zeta_{H}^{2}\\ \widetilde{\zeta}_{A,opt}\ =\ \zeta_{A,opt}+\ \left(0.13+0.12\ \mu+0.4\ \mu^{2} \right)\ \zeta_{H}-\left(0.01+0.9\ \mu+3\ \mu^{2}\right)\ \zeta_{H}^{2}\end{array} \tag{15}\]