# Experimental Testing of Passive Linear TMD for Postural Tremor Attenuation
## 3 Results
### TMD Added to the System  
Measured signals for the response of the experimental arm are presented in the time and frequency domains, before and after the addition of the TMDs. The capability of the TMDs in absorbing the system's vibration is analyzed in terms of the reduction in the amplitude of the critical peak. The reduction is calculated as a percentage using:  
\[\text{Reduction }(\%)=\frac{\text{{Peak}}_{no\ TMD}-\text{{Peak}}_{with\ TMD} \times 100, \tag{1}\]  
where \(\text{{Peak}}_{no\ TMD}\) and \(\text{{Peak}}_{with\ TMD}\) are, respectively, the amplitude of the critical peak between 4-12 Hz, for the response of the system, without and with the TMD(s). The time-domain signal of the experimental arm, excited by the PT signal, is oscillating between \(\pm\)0.1 V as shown in Figure 5. In the frequency domain, the amplitude of the critical peak for the PSD of the signal is 1.069 \(\times\) 10\({}^{-3}\) V\({}^{2}\)/Hz for the principle system, which occurs \begin{table}
\begin{tabular}{c c c c} \hline \hline \(a\) [mm] & \(f_{a}^{meas}\) [Hz] & \(\xi_{a}^{meas}\) [\%] \\ \hline \hline \multirow{2}{*}{**Measurement**} & \multirow{2}{*}{**Graphically**} & **Half-Power Bandwidth** & **Continuous Wavelet** \\  & & **Method** & **Method** \\ \hline
66.0 & 7.57 & 0.32 & 0.20 \\
68.0 & 7.42 & 0.38 & 0.20 \\
70.0 & 7.26 & 0.39 & 0.22 \\
72.0 & 6.95 & 0.36 & 0.21 \\
73.0 & 6.80 & 0.32 & 0.21 \\
74.0 & 6.66 & 0.38 & 0.22 \\
75.0 & 6.55 & 0.38 & 0.21 \\
76.0 & 6.44 & 0.27 & 0.20 \\
77.0 & 6.31 & 0.34 & 0.20 \\
78.0 & 6.17 & 0.26 & 0.17 \\
79.0 & 6.06 & 0.36 & 0.23 \\
80.0 & 5.96 & 0.33 & 0.23 \\
81.0 & 5.83 & 0.26 & 0.20 \\
82.0 & 5.74 & 0.32 & 0.21 \\
83.0 & 5.67 & 0.29 & 0.22 \\
84.0 & 5.52 & 0.27 & 0.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fundamental frequency of the TMD for each screw position and the corresponding modal damping ratio obtained using different methods. at 6.64 Hz. The response of the system is analyzed due to the addition of one TMD designed at the tuning condition, i.e., the frequency of the TMD is chosen to be the critical frequency of the experimental arm (6.64 Hz). A frequency of 6.63 Hz is reached by one TMD system for the screw placed at \(a=74.5\) mm. The behavior of the system before and after the addition of this TMD is represented in Figure 5. The attenuation caused by the tuned absorber is clearly shown in the measured voltage signal (proportional to the linear velocity signal in the direction of motion) of the system in the time domain (Figure 5). The passive absorber shows its ability to operate in reducing the amplitude of the system excited by the PT signal. In the frequency domain (Figure 5), the amplitude at the critical frequency at 6.64 Hz is almost reduced, but the TMD also causes an amplification at a frequency of 8.30 Hz for the coupled system. The reduction is 93% between the critical amplitude of the system without TMD, at 6.64 Hz, and the amplified amplitude, at 8.30 Hz, after the addition of the TMD. Measurements are repeated for the positions of the proof mass presented in Table 1 in order to search for the best position that causes the highest reduction in the amplitude. The responses in Figure 6a are analyzed in the bandwidth of 4 Hz to 12 Hz. No perfect response is obtained, however, a proof mass position can be chosen which allows us to compromise between reduced and amplified amplitudes. To facilitate the analysis, just responses that correspond to the screw placed around the best position are presented. The response of the system is improved as the position of the screw moves from \(a=82.0\) mm towards \(a=76.0\) mm. The highest reduction in the amplitude of the system is observed as the position of the mass is slightly decreased below \(a=76.0\) mm. A position below 74.0 mm deteriorates the response. Therefore, the best position of the mass is between 74.0-76.0 mm approaching 76.0 mm, which corresponds to an operating frequency approximately between 6.44-6.66 Hz.