## 4 Numerical Studies on the Effectiveness of SA-TLGDs  
### SDOF-Wind Turbine with Single SA-TLGD Attached  
The effectiveness of a single SA-TLGD regarding vibration reduction is demonstrated by considering a slender vibration-prone wind turbine with a flat gravity basement (Figure 4a) under severe wind-induced forcing by a time harmonic excitation force \(F(t)\). The wind turbine is modeled as a clamped continuous Bernoulli-Euler beam with the height \(h\) and the head mass \(m_{P}\) on top (Figure 4b). Further \(\rho A\) and \(EI\) denote the tower mass per unit length as well as the bending stiffness. The mass \(m_{P}\) is composed of the mass of the rotor blades, the hub, the nacelle including all internals, and the dead weight of the installed SA-TLGD. The first natural bending frequency \(f_{1}\) with its corresponding mode shape \(\phi_{1}\) is selected as most critical resonance frequency regarding wind-induced vortex shedding, and hence the continuous beam model with an infinite number of DOF can be reduced to a simple SDOF system. The position of the installed single SA-TLGD is assumed at hub height \(h\). The damping ratio of the wind turbine consists of structural damping and aerodynamic damping. Light modal damping is assumed with \(\zeta_{S}=1.4\%\) according to [44] for steel towers. The parameters chosen in this numerical study for the considered wind turbine are listed in Table 1. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter of the Wind Turbine** & **Variable** & **Value** & **Unit** \\ \hline Hub height & \(h\) & 60 & m \\ Tower diameter & \(D\) & 5.5 & m \\ Tower mass per unit length & \(\rho A\) & 3330 & kg/m \\ Bending stiffness & \(EI\) & \(2.9\times 10^{11}\) & Nm\({}^{2}\) \\ Head mass & \(m_{P}\) & 300,000 & kg \\ Modal damping ratio & \(\zeta_{S}\) & 1.4 & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters chosen for the considered wind turbine. By applying the single-unit Ritz approach for the approximated solution of the dynamic deflection along the height of the considered wind turbine tower,  
\[w(x,t)\ =\ q(t)\cdot\phi_{1}(x), \tag{33}\]  
with the properly chosen shape function for the relevant fundamental vibration mode (Figure 2b), which fulfills the kinematic boundary conditions,  
\[\phi_{1}(x)=1-\cos\frac{\pi x}{2h}, \tag{34}\]  
yield the following formulas for the (kinetic equivalent) modal mass and stiffness of the SDOF-main system just from an energy comparison to the equivalent single-mass oscillator [39],  
\[m_{S}=0.227\ \rho Ah+m_{P},\ k_{S}=\frac{\pi^{4}EI}{32\ h^{3}}. \tag{35}\]  
Subsequently, the fundamental circular natural frequency of the SDOF-main system can be written as follows,  
\[\omega_{1}\ =\ \sqrt{\ \frac{k_{S}}{m_{S}}}\ =\ \ \frac{\pi^{2}}{h^{2}}\ \sqrt{\ \frac{EI}{32\ \cdot(0.227\rho A+m_{P}/h)}}. \tag{36}\]  
Inserting the assumed wind turbine parameters listed in Table 1 into Equation (36) results in the circular natural frequency of the considered first vibration mode of the wind turbine to \(\omega_{1}=3.44\) rad/s (\(f_{1}=\omega_{1}/2\pi=0.55\) Hz). The comparison with the measured fundamental frequencies of existing onshore wind turbines shows a very good agreement [45; 46]. While Equation (36) is simple to use, it is noted that it does not consider the flexibility of the foundation and stiffness softening effects due to the axial vertical load arising from the mass \(m_{P}\). The influence of these in practice relevant effects on the resulting natural frequencies of wind turbines was studied in [45; 46]. Optimal tuning of the single SA-TLGD with respect to the relevant fundamental frequency of the force excited SDOF-main system, \(f_{1}=0.55\) Hz, is performed by the  
Figure 4: Wind turbine considered in the numerical study: (**a**) principial drawing of a wind turbine with a flat gravity basement, \(C_{g}\) is the center of gravity; (**b**) mechanical model of the harmonic force excited wind turbine with a single SA-TLGD attached. application of [PERSON]'s formulas (Section 3.1).